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rbeezer  committed fc908ba

Mass text macro replacement in align environments

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File src/section-B.xml

 <alignmath>
 \innerproduct{\vect{v}_i}{\vect{w}}
 <![CDATA[&=\innerproduct{\vect{v}_i}{\sum_{k=1}^{p}a_k\vect{v}_k}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="VRRB" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="VRRB" />\\
 <![CDATA[&=\sum_{k=1}^{p}\innerproduct{\vect{v}_i}{a_k\vect{v}_k}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="IPVA" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="IPVA" />\\
 <![CDATA[&=\sum_{k=1}^{p}a_k\innerproduct{\vect{v}_i}{\vect{v}_k}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="IPSM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="IPSM" />\\
 <![CDATA[&=a_i\innerproduct{\vect{v}_i}{\vect{v}_i}+]]>
 \sum_{\substack{k=1\\k\neq i}}^{p}a_k\innerproduct{\vect{v}_i}{\vect{v}_k}
-<![CDATA[&&]]>\text{<acroref type="property" acro="C" />}\\
+<![CDATA[&&]]><acroref type="property" acro="C" />\\
 <![CDATA[&=a_i(1)+\sum_{\substack{k=1\\k\neq i}}^{p}a_k(0)]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ONS" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ONS" />\\
 <![CDATA[&=a_i]]>
 </alignmath>
 </p>
 <alignmath>
 \adjoint{A}A\vect{y}
 <![CDATA[&=\sum_{i=1}^{n}\innerproduct{\vect{x}_i}{\adjoint{A}A\vect{y}}\vect{x}_i]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="COB" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="COB" />\\
 <![CDATA[&=\sum_{i=1}^{n}\innerproduct{\vect{x}_i}{\adjoint{A}A\sum_{j=1}^{n}a_j\vect{x}_j}\vect{x}_i]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="SSVS" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="SSVS" />\\
 <![CDATA[&=\sum_{i=1}^{n}\innerproduct{\vect{x}_i}{\sum_{j=1}^{n}\adjoint{A}Aa_j\vect{x}_j}\vect{x}_i]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMDAA" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MMDAA" />\\
 <![CDATA[&=\sum_{i=1}^{n}\innerproduct{\vect{x}_i}{\sum_{j=1}^{n}a_j\adjoint{A}A\vect{x}_j}\vect{x}_i]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMSMM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MMSMM" />\\
 <![CDATA[&=\sum_{i=1}^{n}\sum_{j=1}^{n}\innerproduct{\vect{x}_i}{a_j\adjoint{A}A\vect{x}_j}\vect{x}_i]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="IPVA" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="IPVA" />\\
 <![CDATA[&=\sum_{i=1}^{n}\sum_{j=1}^{n}a_j\innerproduct{\vect{x}_i}{\adjoint{A}A\vect{x}_j}\vect{x}_i]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="IPSM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="IPSM" />\\
 <![CDATA[&=\sum_{i=1}^{n}\sum_{j=1}^{n}a_j\innerproduct{A\vect{x}_i}{A\vect{x}_j}\vect{x}_i]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="AIP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="AIP" />\\
 <![CDATA[&=]]>
 \sum_{i=1}^{n}\sum_{\substack{j=1\\j\neq i}}^{n}a_j\innerproduct{A\vect{x}_i}{A\vect{x}_j}\vect{x}_i
 +
 \sum_{\ell=1}^{n}a_\ell\innerproduct{A\vect{x}_\ell}{A\vect{x}_\ell}\vect{x}_\ell
-<![CDATA[&&]]>\text{<acroref type="property" acro="C" />}\\
+<![CDATA[&&]]><acroref type="property" acro="C" />\\
 <![CDATA[&=]]>
 \sum_{i=1}^{n}\sum_{\substack{j=1\\j\neq i}}^{n}a_j(0)\vect{x}_i
 +
 \sum_{\ell=1}^{n}a_\ell(1)\vect{x}_\ell
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ONS" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ONS" />\\
 <![CDATA[&=]]>
 \sum_{i=1}^{n}\sum_{\substack{j=1\\j\neq i}}^{n}\zerovector
 +
 \sum_{\ell=1}^{n}a_\ell\vect{x}_\ell
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="ZSSM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="ZSSM" />\\
 <![CDATA[&=\sum_{\ell=1}^{n}a_\ell\vect{x}_\ell]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="Z" />}\\
+<![CDATA[&&]]><acroref type="property" acro="Z" />\\
 <![CDATA[&=\vect{y}\\]]>
 <![CDATA[&=I_n\vect{y}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMIM" />}
+<![CDATA[&&]]><acroref type="theorem" acro="MMIM" />
 </alignmath>
 </p>
 
 Now consider the entry in row $i$ and column $j$ for these equal matrices,
 <alignmath>
 0
-<![CDATA[&=\matrixentry{\zeromatrix}{ij}&&]]>\text{<acroref type="definition" acro="ZM" />}\\
+<![CDATA[&=\matrixentry{\zeromatrix}{ij}&&]]><acroref type="definition" acro="ZM" />\\
 <![CDATA[&=\matrixentry{\sum_{k=1}^{m}\sum_{\ell=1}^{n}\alpha_{k\ell}B_{k\ell}}{ij}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ME" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ME" />\\
 <![CDATA[&=\sum_{k=1}^{m}\sum_{\ell=1}^{n}\matrixentry{\alpha_{k\ell}B_{k\ell}}{ij}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="MA" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="MA" />\\
 <![CDATA[&=\sum_{k=1}^{m}\sum_{\ell=1}^{n}\alpha_{k\ell}\matrixentry{B_{k\ell}}{ij}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="MSM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="MSM" />\\
 <![CDATA[&=\alpha_{ij}\matrixentry{B_{ij}}{ij}&&]]>
 \text{$\matrixentry{B_{k\ell}}{ij}=0$ when $(k,\ell)\neq(i,j)$}\\
 <![CDATA[&=\alpha_{ij}(1)&&\text{$\matrixentry{B_{ij}}{ij}=1$}\\]]>
 <alignmath>
 \matrixentry{C}{ij}
 <![CDATA[&=\matrixentry{\sum_{k=1}^{m}\sum_{\ell=1}^{n}\matrixentry{A}{k\ell}B_{k\ell}}{ij}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ME" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ME" />\\
 <![CDATA[&=\sum_{k=1}^{m}\sum_{\ell=1}^{n}\matrixentry{\matrixentry{A}{k\ell}B_{k\ell}}{ij}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="MA" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="MA" />\\
 <![CDATA[&=\sum_{k=1}^{m}\sum_{\ell=1}^{n}\matrixentry{A}{k\ell}\matrixentry{B_{k\ell}}{ij}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="MSM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="MSM" />\\
 <![CDATA[&=\matrixentry{A}{ij}\matrixentry{B_{ij}}{ij}]]>
 <![CDATA[&&\text{$\matrixentry{B_{k\ell}}{ij}=0$ when $(k,\ell)\neq(i,j)$}\\]]>
 <![CDATA[&=\matrixentry{A}{ij}(1)&&\text{$\matrixentry{B_{ij}}{ij}=1$}\\]]>
 a_3A\vect{x}_3+
 \cdots+
 a_nA\vect{x}_n
-<![CDATA[&&]]>\text{<acroref type="definition" acro="RLD" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="RLD" />\\
 <![CDATA[&=]]>
 Aa_1\vect{x}_1+
 Aa_2\vect{x}_2+
 Aa_3\vect{x}_3+
 \cdots+
 Aa_n\vect{x}_n
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMSMM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MMSMM" />\\
 <![CDATA[&=]]>
 A\left(
 a_1\vect{x}_1+
 \cdots+
 a_n\vect{x}_n
 \right)
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMDAA" />}
+<![CDATA[&&]]><acroref type="theorem" acro="MMDAA" />
 </alignmath>
 Since $A$ is nonsingular, <acroref type="definition" acro="NM" /> and <acroref type="theorem" acro="SLEMM" /> allows us to conclude that
 <alignmath>
 <alignmath>
 \vect{y}
 <![CDATA[&=A\vect{w}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="SLEMM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="SLEMM" />\\
 <![CDATA[&=A\left(\lincombo{b}{x}{n}\right)]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="SSVS" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="SSVS" />\\
 <![CDATA[&=]]>
 Ab_1\vect{x}_1+
 Ab_2\vect{x}_2+
 Ab_3\vect{x}_3+
 \cdots+
 Ab_n\vect{x}_n
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMDAA" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MMDAA" />\\
 <![CDATA[&=]]>
 b_1A\vect{x}_1+
 b_2A\vect{x}_2+
 b_3A\vect{x}_3+
 \cdots+
 b_nA\vect{x}_n
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMSMM" />}
+<![CDATA[&&]]><acroref type="theorem" acro="MMSMM" />
 </alignmath>
 So we can write an arbitrary vector of $\complex{n}$ as a linear combination of the elements of $C$.  In other words, $C$ spans $\complex{n}$ (<acroref type="definition" acro="SSVS" />).  By <acroref type="definition" acro="B" />, the set $C$ is a basis for $\complex{n}$.<br /><br />
 <implyreverse /> Assume that $C$ is a basis and prove that $A$ is nonsingular.  Let $\vect{x}$ be a solution to the homogeneous system $\homosystem{A}$.  Since $B$ is a basis of $\complex{n}$ there are  scalars, $\scalarlist{a}{n}$, such that
 <alignmath>
 \zerovector
 <![CDATA[&=A\vect{x}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="SLEMM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="SLEMM" />\\
 <![CDATA[&=A\left(\lincombo{a}{x}{n}\right)]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="SSVS" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="SSVS" />\\
 <![CDATA[&=]]>
 Aa_1\vect{x}_1+
 Aa_2\vect{x}_2+
 Aa_3\vect{x}_3+
 \cdots+
 Aa_n\vect{x}_n
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMDAA" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MMDAA" />\\
 <![CDATA[&=]]>
 a_1A\vect{x}_1+
 a_2A\vect{x}_2+
 a_3A\vect{x}_3+
 \cdots+
 a_nA\vect{x}_n
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMSMM" />}
+<![CDATA[&&]]><acroref type="theorem" acro="MMSMM" />
 </alignmath>
 This is a relation of linear dependence on the linearly independent set $C$, so the scalars must all be zero, $a_1=a_2=a_3=\cdots=a_n=0$.  Thus,
 <alignmath>
 <alignmath>
 A\text{ nonsingular}
 <![CDATA[&\iff AG\text{ nonsingular}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="NPNT" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="NPNT" />\\
 <![CDATA[&\iff C\text{ basis for }\complex{n}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="CNMB" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="CNMB" />\\
 </alignmath>
 That was easy!
 </solution>

File src/section-CB.xml

 <alignmath>
 \vectrep{C}{\vect{v}}
 <![CDATA[&=\vectrep{C}{\lt{I_V}{\vect{v}}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="IDLT" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="IDLT" />\\
 <![CDATA[&=\matrixrep{I_V}{B}{C}\vectrep{B}{\vect{v}}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="FTMR" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="FTMR" />\\
 <![CDATA[&=\cbm{B}{C}\vectrep{B}{\vect{v}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="CBM" />}
+<![CDATA[&&]]><acroref type="definition" acro="CBM" />
 </alignmath>
 </p>
 
 <p>Then
 <alignmath>
 \inverse{\cbm{B}{C}}
-<![CDATA[&=\inverse{\left(\matrixrep{I_V}{B}{C}\right)}&&]]>\text{<acroref type="definition" acro="CBM" />}\\
-<![CDATA[&=\matrixrep{\ltinverse{I_V}}{C}{B}&&]]>\text{<acroref type="theorem" acro="IMR" />}\\
-<![CDATA[&=\matrixrep{I_V}{C}{B}&&]]>\text{<acroref type="definition" acro="IDLT" />}\\
-<![CDATA[&=\cbm{C}{B}&&]]>\text{<acroref type="definition" acro="CBM" />}\\
+<![CDATA[&=\inverse{\left(\matrixrep{I_V}{B}{C}\right)}&&]]><acroref type="definition" acro="CBM" />\\
+<![CDATA[&=\matrixrep{\ltinverse{I_V}}{C}{B}&&]]><acroref type="theorem" acro="IMR" />\\
+<![CDATA[&=\matrixrep{I_V}{C}{B}&&]]><acroref type="definition" acro="IDLT" />\\
+<![CDATA[&=\cbm{C}{B}&&]]><acroref type="definition" acro="CBM" />\\
 </alignmath>
 </p>
 
 <p>Applying <acroref type="theorem" acro="CB" />, we obtain a second representation of $\vect{u}$, but now relative to $C$,
 <alignmath>
 \vectrep{C}{\vect{u}}
-<![CDATA[&=\cbm{B}{C}\vectrep{B}{\vect{u}}&&]]>\text{<acroref type="theorem" acro="CB" />}\\
+<![CDATA[&=\cbm{B}{C}\vectrep{B}{\vect{u}}&&]]><acroref type="theorem" acro="CB" />\\
 <![CDATA[&=]]>
 \begin{bmatrix}
 <![CDATA[1 &-1 & 0 & 0 & 0\\]]>
 <![CDATA[0 & 0 & 0 & 0 & 1\\]]>
 \end{bmatrix}
 \colvector{5\\-3\\2\\8\\-3}\\
-<![CDATA[&=\colvector{8\\-5\\-6\\11\\-3}&&]]>\text{<acroref type="definition" acro="MVP" />}
+<![CDATA[&=\colvector{8\\-5\\-6\\11\\-3}&&]]><acroref type="definition" acro="MVP" />
 </alignmath>
 </p>
 
 <alignmath>
 \vect{u}
 <![CDATA[&=\vectrepinv{C}{\vectrep{C}{\vect{u}}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="IVLT" />\\
 <![CDATA[&=\vectrepinv{C}{\colvector{8\\-5\\-6\\11\\-3}}\\]]>
 <![CDATA[&=8(1)+(-5)(1+x)+(-6)(1+x+x^2)\\]]>
 <![CDATA[&\quad\quad+(11)(1+x+x^2+x^3)+(-3)(1+x+x^2+x^3+x^4)]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="VR" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="VR" />\\
 <![CDATA[&=5-3x+2x^2+8x^3-3x^4]]>
 </alignmath>
 </p>
 <p>We formed two representations of the vector $\vect{u}$ above, so we can again provide a check on our computations by converting from the representation of $\vect{u}$ relative to $C$ to the representation of $\vect{u}$ relative to $B$,
 <alignmath>
 \vectrep{B}{\vect{u}}
-<![CDATA[&=\cbm{C}{B}\vectrep{C}{\vect{u}}&&]]>\text{<acroref type="theorem" acro="CB" />}\\
+<![CDATA[&=\cbm{C}{B}\vectrep{C}{\vect{u}}&&]]><acroref type="theorem" acro="CB" />\\
 <![CDATA[&=]]>
 \begin{bmatrix}
 <![CDATA[1 & 1 & 1 & 1 & 1\\]]>
 <![CDATA[0 & 0 & 0 & 0 & 1\\]]>
 \end{bmatrix}
 \colvector{8\\-5\\-6\\11\\-3}\\
-<![CDATA[&=\colvector{5\\-3\\2\\8\\-3}&&]]>\text{<acroref type="definition" acro="MVP" />}\\
+<![CDATA[&=\colvector{5\\-3\\2\\8\\-3}&&]]><acroref type="definition" acro="MVP" />\\
 </alignmath>
 </p>
 
 <p>Now, applying <acroref type="theorem" acro="CB" /> we can convert the representation of $\vect{y}$ relative to $B$ into a representation relative to $C$,
 <alignmath>
 \vectrep{C}{\vect{y}}
-<![CDATA[&=\cbm{B}{C}\vectrep{B}{\vect{y}}&&]]>\text{<acroref type="theorem" acro="CB" />}\\
+<![CDATA[&=\cbm{B}{C}\vectrep{B}{\vect{y}}&&]]><acroref type="theorem" acro="CB" />\\
 <![CDATA[&=]]>
 \begin{bmatrix}
 <![CDATA[1 & 2 & 1 & 2 \\]]>
 <![CDATA[-1 & 0 & -2 & 3]]>
 \end{bmatrix}
 \colvector{-21\\6\\11\\-7}\\
-<![CDATA[&=\colvector{-12\\5\\-20\\-22}&&]]>\text{<acroref type="definition" acro="MVP" />}
+<![CDATA[&=\colvector{-12\\5\\-20\\-22}&&]]><acroref type="definition" acro="MVP" />
 </alignmath>
 </p>
 
 <p>
 <alignmath>
 \cbm{E}{D}\matrixrep{T}{C}{E}\cbm{B}{C}
-<![CDATA[&=\matrixrep{I_V}{E}{D}\matrixrep{T}{C}{E}\matrixrep{I_U}{B}{C}&&]]>\text{<acroref type="definition" acro="CBM" />}\\
-<![CDATA[&=\matrixrep{I_V}{E}{D}\matrixrep{\compose{T}{I_U}}{B}{E}&&]]>\text{<acroref type="theorem" acro="MRCLT" />}\\
-<![CDATA[&=\matrixrep{I_V}{E}{D}\matrixrep{T}{B}{E}&&]]>\text{<acroref type="definition" acro="IDLT" />}\\
-<![CDATA[&=\matrixrep{\compose{I_V}{T}}{B}{D}&&]]>\text{<acroref type="theorem" acro="MRCLT" />}\\
-<![CDATA[&=\matrixrep{T}{B}{D}&&]]>\text{<acroref type="definition" acro="IDLT" />}
+<![CDATA[&=\matrixrep{I_V}{E}{D}\matrixrep{T}{C}{E}\matrixrep{I_U}{B}{C}&&]]><acroref type="definition" acro="CBM" />\\
+<![CDATA[&=\matrixrep{I_V}{E}{D}\matrixrep{\compose{T}{I_U}}{B}{E}&&]]><acroref type="theorem" acro="MRCLT" />\\
+<![CDATA[&=\matrixrep{I_V}{E}{D}\matrixrep{T}{B}{E}&&]]><acroref type="definition" acro="IDLT" />\\
+<![CDATA[&=\matrixrep{\compose{I_V}{T}}{B}{D}&&]]><acroref type="theorem" acro="MRCLT" />\\
+<![CDATA[&=\matrixrep{T}{B}{D}&&]]><acroref type="definition" acro="IDLT" />
 </alignmath>
 </p>
 
 <alignmath>
 \cbm{E}{D}
 <![CDATA[&=\inverse{\left(\cbm{D}{E}\right)}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="ICBM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="ICBM" />\\
 <![CDATA[&=\inverse{]]>
 \begin{bmatrix}
 <![CDATA[ 2 & -1 & -3 & 0 \\]]>
 <alignmath>
 \matrixrep{Q}{B}{D}
 <![CDATA[&=\cbm{E}{D}\matrixrep{Q}{C}{E}\cbm{B}{C}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MRCB" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MRCB" />\\
 <![CDATA[&=]]>
 \begin{bmatrix}
 <![CDATA[ 1 & -2 & 1 & 1 \\]]>
 <p>In the conclusion of <acroref type="theorem" acro="MRCB" />, replace $D$ by $B$, and replace $E$ by $C$,
 <alignmath>
 \matrixrep{T}{B}{B}
-<![CDATA[&=\cbm{C}{B}\matrixrep{T}{C}{C}\cbm{B}{C}&&]]>\text{<acroref type="theorem" acro="MRCB" />}\\
-<![CDATA[&=\inverse{\cbm{B}{C}}\matrixrep{T}{C}{C}\cbm{B}{C}&&]]>\text{<acroref type="theorem" acro="ICBM" />}
+<![CDATA[&=\cbm{C}{B}\matrixrep{T}{C}{C}\cbm{B}{C}&&]]><acroref type="theorem" acro="MRCB" />\\
+<![CDATA[&=\inverse{\cbm{B}{C}}\matrixrep{T}{C}{C}\cbm{B}{C}&&]]><acroref type="theorem" acro="ICBM" />
 </alignmath>
 </p>
 
 <p><implyforward /> Assume that $\vect{v}\in V$ is an eigenvector of $T$ for the eigenvalue $\lambda$.  Then
 <alignmath>
 \matrixrep{T}{B}{B}\vectrep{B}{\vect{v}}
-<![CDATA[&=\vectrep{B}{\lt{T}{\vect{v}}}&&]]>\text{<acroref type="theorem" acro="FTMR" />}\\
-<![CDATA[&=\vectrep{B}{\lambda\vect{v}}&&]]>\text{<acroref type="definition" acro="EELT" />}\\
-<![CDATA[&=\lambda\vectrep{B}{\vect{v}}&&]]>\text{<acroref type="theorem" acro="VRLT" />}
+<![CDATA[&=\vectrep{B}{\lt{T}{\vect{v}}}&&]]><acroref type="theorem" acro="FTMR" />\\
+<![CDATA[&=\vectrep{B}{\lambda\vect{v}}&&]]><acroref type="definition" acro="EELT" />\\
+<![CDATA[&=\lambda\vectrep{B}{\vect{v}}&&]]><acroref type="theorem" acro="VRLT" />
 </alignmath>
 which by <acroref type="definition" acro="EEM" /> says that $\vectrep{B}{\vect{v}}$ is an eigenvector of the matrix $\matrixrep{T}{B}{B}$ for the eigenvalue $\lambda$.</p>
 
 <p><implyreverse />  Assume that $\vectrep{B}{\vect{v}}$ is an eigenvector of $\matrixrep{T}{B}{B}$ for the eigenvalue $\lambda$.  Then
 <alignmath>
 \lt{T}{\vect{v}}
-<![CDATA[&=\vectrepinv{B}{\vectrep{B}{\lt{T}{\vect{v}}}}&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
-<![CDATA[&=\vectrepinv{B}{\matrixrep{T}{B}{B}\vectrep{B}{\vect{v}}}&&]]>\text{<acroref type="theorem" acro="FTMR" />}\\
-<![CDATA[&=\vectrepinv{B}{\lambda\vectrep{B}{\vect{v}}}&&]]>\text{<acroref type="definition" acro="EEM" />}\\
-<![CDATA[&=\lambda\vectrepinv{B}{\vectrep{B}{\vect{v}}}&&]]>\text{<acroref type="theorem" acro="ILTLT" />}\\
-<![CDATA[&=\lambda\vect{v}&&]]>\text{<acroref type="definition" acro="IVLT" />}
+<![CDATA[&=\vectrepinv{B}{\vectrep{B}{\lt{T}{\vect{v}}}}&&]]><acroref type="definition" acro="IVLT" />\\
+<![CDATA[&=\vectrepinv{B}{\matrixrep{T}{B}{B}\vectrep{B}{\vect{v}}}&&]]><acroref type="theorem" acro="FTMR" />\\
+<![CDATA[&=\vectrepinv{B}{\lambda\vectrep{B}{\vect{v}}}&&]]><acroref type="definition" acro="EEM" />\\
+<![CDATA[&=\lambda\vectrepinv{B}{\vectrep{B}{\vect{v}}}&&]]><acroref type="theorem" acro="ILTLT" />\\
+<![CDATA[&=\lambda\vect{v}&&]]><acroref type="definition" acro="IVLT" />
 </alignmath>
 which by <acroref type="definition" acro="EELT" /> says $\vect{v}$ is an eigenvector of $T$ for the eigenvalue $\lambda$.</p>
 
 <alignmath>
 <![CDATA[\lt{\ltinverse{T}}{\vect{v}}&=]]>
 <![CDATA[\frac{1}{\lambda}\lambda\lt{\ltinverse{T}}{\vect{v}}&&\text{$\lambda\neq 0$}\\]]>
-<![CDATA[&=\frac{1}{\lambda}\lt{\ltinverse{T}}{\lambda\vect{v}}&&]]>\text{<acroref type="theorem" acro="ILTLT" />}\\
+<![CDATA[&=\frac{1}{\lambda}\lt{\ltinverse{T}}{\lambda\vect{v}}&&]]><acroref type="theorem" acro="ILTLT" />\\
 <![CDATA[&=\frac{1}{\lambda}\lt{\ltinverse{T}}{\lt{T}{\vect{v}}}&&\text{$\vect{v}$ eigenvector of $T$}\\]]>
-<![CDATA[&=\frac{1}{\lambda}\lt{I_V}{\vect{v}}&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
-<![CDATA[&=\frac{1}{\lambda}\vect{v}&&]]>\text{<acroref type="definition" acro="IDLT" />}
+<![CDATA[&=\frac{1}{\lambda}\lt{I_V}{\vect{v}}&&]]><acroref type="definition" acro="IVLT" />\\
+<![CDATA[&=\frac{1}{\lambda}\vect{v}&&]]><acroref type="definition" acro="IDLT" />
 </alignmath>
 which says that $\displaystyle\frac{1}{\lambda}$ is an eigenvalue of $\ltinverse{T}$ with eigenvector $\vect{v}$.  Note that it is possible to prove that any eigenvalue of an invertible linear transformation is never zero.  So the hypothesis that $\lambda$ be nonzero is just a convenience for this problem.
 </solution>

File src/section-CRS.xml

 <alignmath>
 AB
 <![CDATA[&=A\matrixcolumns{b}{n}\\]]>
-<![CDATA[&=[A\vect{b}_1|A\vect{b}_2|A\vect{b}_3|\ldots|A\vect{b}_n]&&]]>\text{<acroref type="definition" acro="MM" />}\\
+<![CDATA[&=[A\vect{b}_1|A\vect{b}_2|A\vect{b}_3|\ldots|A\vect{b}_n]&&]]><acroref type="definition" acro="MM" />\\
 <![CDATA[&=\matrixcolumns{e}{n}\\]]>
-<![CDATA[&=I_n&&]]>\text{<acroref type="definition" acro="SUV" />}\\
+<![CDATA[&=I_n&&]]><acroref type="definition" acro="SUV" />\\
 </alignmath></p>
 
 <p>So the matrix $B$ is a <q>right-inverse</q> for $A$.  By <acroref type="theorem" acro="NMRRI" />, $I_n$ is a nonsingular matrix, so by <acroref type="theorem" acro="NPNT" /> both $A$ and $B$ are nonsingular.  Thus, in particular, $A$ is nonsingular.  (<contributorname code="travisosborne" /> contributed to this proof.)</p>
 <p>Now consider this vector equality in location $d_i$.  Since $B$ is in reduced row-echelon form, the entries of column $d_i$ of $B$ are all zero, except for a (leading) 1 in row $i$.  Thus, in $\transpose{B}$, row $d_i$ is all zeros, excepting a 1 in column $i$.   So, for $1\leq i\leq r$,
 <alignmath>
 0
-<![CDATA[&=\vectorentry{\zerovector}{d_i}&&]]>\text{<acroref type="definition" acro="ZCV" />}\\
+<![CDATA[&=\vectorentry{\zerovector}{d_i}&&]]><acroref type="definition" acro="ZCV" />\\
 <![CDATA[&=\vectorentry{\lincombo{\alpha}{B}{r}}{d_i}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="RLDCV" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="RLDCV" />\\
 <![CDATA[&=]]>
 \vectorentry{\alpha_1\vect{B}_1}{d_i}+
 \vectorentry{\alpha_2\vect{B}_2}{d_i}+
 \vectorentry{\alpha_3\vect{B}_3}{d_i}+
 \cdots+
 \vectorentry{\alpha_r\vect{B}_r}{d_i}+
-<![CDATA[&&]]>\text{<acroref type="definition" acro="MA" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="MA" />\\
 <![CDATA[&=]]>
 \alpha_1\vectorentry{\vect{B}_1}{d_i}+
 \alpha_2\vectorentry{\vect{B}_2}{d_i}+
 \alpha_3\vectorentry{\vect{B}_3}{d_i}+
 \cdots+
 \alpha_r\vectorentry{\vect{B}_r}{d_i}+
-<![CDATA[&&]]>\text{<acroref type="definition" acro="MSM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="MSM" />\\
 <![CDATA[&=]]>
 \alpha_1(0)+
 \alpha_2(0)+
 \alpha_i(1)+
 \cdots+
 \alpha_r(0)
-<![CDATA[&&]]>\text{<acroref type="definition" acro="RREF" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="RREF" />\\
 <![CDATA[&=\alpha_i]]>
 </alignmath>
 </p>
 <p><alignmath>
 \csp{A}
 <![CDATA[&=\csp{\transpose{\left(\transpose{A}\right)}}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="TT" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="TT" />\\
 <![CDATA[&=\rsp{\transpose{A}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="RSM" />}
+<![CDATA[&&]]><acroref type="definition" acro="RSM" />
 </alignmath>
 </p>
 
 </problem>
 <solution contributor="robertbeezer"><alignmath>
 \vect{y}\in\rsp{A}
-<![CDATA[&\iff\vect{y}\in\csp{\transpose{A}}&&]]>\text{<acroref type="definition" acro="RSM" />}\\
-<![CDATA[&\iff\linearsystem{\transpose{A}}{\vect{y}}\text{ is consistent}&&]]>\text{<acroref type="theorem" acro="CSCS" />}
+<![CDATA[&\iff\vect{y}\in\csp{\transpose{A}}&&]]><acroref type="definition" acro="RSM" />\\
+<![CDATA[&\iff\linearsystem{\transpose{A}}{\vect{y}}\text{ is consistent}&&]]><acroref type="theorem" acro="CSCS" />
 </alignmath>
 The augmented matrix $\augmented{\transpose{A}}{\vect{y}}$ row reduces to
 <alignmath>
 <alignmath>
 A\vect{y}
 <![CDATA[&=A\left(B\vect{w}\right)&&\text{Definition of $\vect{y}$}\\]]>
-<![CDATA[&=\left(AB\right)\vect{w}&&]]>\text{<acroref type="theorem" acro="MMA" />}\\
+<![CDATA[&=\left(AB\right)\vect{w}&&]]><acroref type="theorem" acro="MMA" />\\
 <![CDATA[&=\vect{x}&&\text{$\vect{w}$ solution to $\linearsystem{AB}{\vect{x}}$}\\]]>
 </alignmath>
 This says that $\linearsystem{A}{\vect{x}}$ is a consistent system, and by <acroref type="theorem" acro="CSCS" />, we see that $\vect{x}\in\csp{A}$ and therefore $\csp{AB}\subseteq\csp{A}$.<br /><br />
 Choose $\vect{x}\in\csp{A}$.  By <acroref type="theorem" acro="CSCS" /> the linear system $\linearsystem{A}{\vect{x}}$ is consistent, so let $\vect{y}$ be one such solution.  Because $B$ is nonsingular, and linear system using $B$ as a coefficient matrix will have a solution (<acroref type="theorem" acro="NMUS" />).  Let $\vect{w}$ be the unique solution to the linear system $\linearsystem{B}{\vect{y}}$.  All set, here we go,
 <alignmath>
 \left(AB\right)\vect{w}
-<![CDATA[&=A\left(B\vect{w}\right)&&]]>\text{<acroref type="theorem" acro="MMA" />}\\
+<![CDATA[&=A\left(B\vect{w}\right)&&]]><acroref type="theorem" acro="MMA" />\\
 <![CDATA[&=A\vect{y}&&\text{$\vect{w}$ solution to $\linearsystem{B}{\vect{y}}$}\\]]>
 <![CDATA[&=\vect{x}&&\text{$\vect{y}$ solution to $\linearsystem{A}{\vect{x}}$}\\]]>
 </alignmath>
 <solution contributor="robertbeezer">First, $\zerovector\in\nsp{B}$ trivially.  Now suppose that $\vect{x}\in\nsp{B}$.  Then
 <alignmath>
 AB\vect{x}
-<![CDATA[&=A(B\vect{x})&&]]>\text{<acroref type="theorem" acro="MMA" />}\\
+<![CDATA[&=A(B\vect{x})&&]]><acroref type="theorem" acro="MMA" />\\
 <![CDATA[&=A\zerovector&&\vect{x}\in\nsp{B}\\]]>
-<![CDATA[&=\zerovector&&]]>\text{<acroref type="theorem" acro="MMZM" />}
+<![CDATA[&=\zerovector&&]]><acroref type="theorem" acro="MMZM" />
 </alignmath>
 Since we have assumed $AB$ is nonsingular, <acroref type="definition" acro="NM" /> implies that $\vect{x}=\zerovector$.<br /><br />
 Second, $\zerovector\in\csp{B}$ and $\zerovector\in\nsp{A}$ trivially, and so the zero vector is in the intersection as well (<acroref type="definition" acro="SI" />).  Now suppose that $\vect{y}\in\csp{B}\cap\nsp{A}$.  Because $\vect{y}\in\csp{B}$, <acroref type="theorem" acro="CSCS" /> says the system $\linearsystem{B}{\vect{y}}$ is consistent.  Let $\vect{x}\in\complex{n}$ be one solution to this system.  Then
 <alignmath>
 AB\vect{x}
-<![CDATA[&=A(B\vect{x})&&]]>\text{<acroref type="theorem" acro="MMA" />}\\
+<![CDATA[&=A(B\vect{x})&&]]><acroref type="theorem" acro="MMA" />\\
 <![CDATA[&=A\vect{y}&&\text{$\vect{x}$ solution to $\linearsystem{B}{\vect{y}}$}\\]]>
 <![CDATA[&=\zerovector&&\vect{y}\in\nsp{A}]]>
 </alignmath>

File src/section-D.xml

 <alignmath>
 <![CDATA[&\lincombo{c}{u}{m}\\]]>
 <![CDATA[&=c_{1}\left(a_{11}\vect{v}_1+a_{21}\vect{v}_2+a_{31}\vect{v}_3+\cdots+a_{t1}\vect{v}_t\right)]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="SSVS" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="SSVS" />\\
 <![CDATA[&\quad\quad+c_{2}\left(a_{12}\vect{v}_1+a_{22}\vect{v}_2+a_{32}\vect{v}_3+\cdots+a_{t2}\vect{v}_t\right)\\]]>
 <![CDATA[&\quad\quad+c_{3}\left(a_{13}\vect{v}_1+a_{23}\vect{v}_2+a_{33}\vect{v}_3+\cdots+a_{t3}\vect{v}_t\right)\\]]>
 <![CDATA[&\quad\quad\quad\quad\vdots\\]]>
 <![CDATA[&\quad\quad+c_{m}\left(a_{1m}\vect{v}_1+a_{2m}\vect{v}_2+a_{3m}\vect{v}_3+\cdots+a_{tm}\vect{v}_t\right)\\]]>
 <![CDATA[&=c_{1}a_{11}\vect{v}_1+c_{1}a_{21}\vect{v}_2+c_{1}a_{31}\vect{v}_3+\cdots+c_{1}a_{t1}\vect{v}_t]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="DVA" />}\\
+<![CDATA[&&]]><acroref type="property" acro="DVA" />\\
 <![CDATA[&\quad\quad+c_{2}a_{12}\vect{v}_1+c_{2}a_{22}\vect{v}_2+c_{2}a_{32}\vect{v}_3+\cdots+c_{2}a_{t2}\vect{v}_t\\]]>
 <![CDATA[&\quad\quad+c_{3}a_{13}\vect{v}_1+c_{3}a_{23}\vect{v}_2+c_{3}a_{33}\vect{v}_3+\cdots+c_{3}a_{t3}\vect{v}_t\\]]>
 <![CDATA[&\quad\quad\quad\quad\vdots\\]]>
 <![CDATA[&\quad\quad+c_{m}a_{1m}\vect{v}_1+c_{m}a_{2m}\vect{v}_2+c_{m}a_{3m}\vect{v}_3+\cdots+c_{m}a_{tm}\vect{v}_t\\]]>
 <![CDATA[&=\left(c_{1}a_{11}+c_{2}a_{12}+c_{3}a_{13}+\cdots+c_{m}a_{1m}\right)\vect{v}_1]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="DSA" />}\\
+<![CDATA[&&]]><acroref type="property" acro="DSA" />\\
 <![CDATA[&\quad\quad+\left(c_{1}a_{21}+c_{2}a_{22}+c_{3}a_{23}+\cdots+c_{m}a_{2m}\right)\vect{v}_2\\]]>
 <![CDATA[&\quad\quad+\left(c_{1}a_{31}+c_{2}a_{32}+c_{3}a_{33}+\cdots+c_{m}a_{3m}\right)\vect{v}_3\\]]>
 <![CDATA[&\quad\quad\quad\quad\vdots\\]]>
 <![CDATA[&\quad\quad+\left(c_{1}a_{t1}+c_{2}a_{t2}+c_{3}a_{t3}+\cdots+c_{m}a_{tm}\right)\vect{v}_t\\]]>
 <![CDATA[&=\left(a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3}+\cdots+a_{1m}c_{m}\right)\vect{v}_1]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="CMCN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CMCN" />\\
 <![CDATA[&\quad\quad+\left(a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3}+\cdots+a_{2m}c_{m}\right)\vect{v}_2\\]]>
 <![CDATA[&\quad\quad+\left(a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}+\cdots+a_{3m}c_{m}\right)\vect{v}_3\\]]>
 <![CDATA[&\quad\quad\quad\quad\vdots\\]]>
 <![CDATA[&=0\vect{v}_1+0\vect{v}_2+0\vect{v}_3+\cdots+0\vect{v}_t]]>
 <![CDATA[&&\text{$c_j$ as solution}\\]]>
 <![CDATA[&=\zerovector+\zerovector+\zerovector+\cdots+\zerovector]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="ZSSM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="ZSSM" />\\
 <![CDATA[&=\zerovector]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="Z" />}
+<![CDATA[&&]]><acroref type="property" acro="Z" />
 </alignmath>
 </p>
 
 <alignmath>
 \sum_{j=1}^{m}c_{j}\vect{u}_j
 <![CDATA[&=\sum_{j=1}^{m}c_{j}\left(\sum_{i=1}^{t}a_{ij}\vect{v_i}\right)]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="SSVS" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="SSVS" />\\
 <![CDATA[&=\sum_{j=1}^{m}\sum_{i=1}^{t}c_{j}a_{ij}\vect{v}_i]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="DVA" />}\\
+<![CDATA[&&]]><acroref type="property" acro="DVA" />\\
 <![CDATA[&=\sum_{i=1}^{t}\sum_{j=1}^{m}c_{j}a_{ij}\vect{v}_i]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="CMCN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CMCN" />\\
 <![CDATA[&=\sum_{i=1}^{t}\sum_{j=1}^{m}a_{ij}c_{j}\vect{v}_i]]>
 <![CDATA[&&\text{Commutativity in $\complex{}$}\\]]>
 <![CDATA[&=\sum_{i=1}^{t}\left(\sum_{j=1}^{m}a_{ij}c_{j}\right)\vect{v}_i]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="DSA" />}\\
+<![CDATA[&&]]><acroref type="property" acro="DSA" />\\
 <![CDATA[&=\sum_{i=1}^{t}0\vect{v}_i]]>
 <![CDATA[&&\text{$c_j$ as solution}\\]]>
 <![CDATA[&=\sum_{i=1}^{t}\zerovector]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="ZSSM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="ZSSM" />\\
 <![CDATA[&=\zerovector]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="Z" />}
+<![CDATA[&&]]><acroref type="property" acro="Z" />
 </alignmath>
 </p>
 
 
 <p>(2 $\Rightarrow$ 3)  Suppose $\rank{A}=n$.  Then <acroref type="theorem" acro="RPNC" /> gives
 <alignmath>
-<![CDATA[\nullity{A}&=n-\rank{A}&&]]>\text{<acroref type="theorem" acro="RPNC" />}\\
+<![CDATA[\nullity{A}&=n-\rank{A}&&]]><acroref type="theorem" acro="RPNC" />\\
 <![CDATA[&=n-n&&\text{Hypothesis}\\]]>
 <![CDATA[&=0]]>
 </alignmath>
 So $r=3$ for this matrix.  Then
 <alignmath>
 \dimension{\nsp{A}}
-<![CDATA[&=\nullity{A}&&]]>\text{<acroref type="definition" acro="NOM" />}\\
+<![CDATA[&=\nullity{A}&&]]><acroref type="definition" acro="NOM" />\\
 <![CDATA[&=\left(\nullity{A}+\rank{A}\right)-\rank{A}\\]]>
-<![CDATA[&=5-\rank{A}&&]]>\text{<acroref type="theorem" acro="RPNC" />}\\
-<![CDATA[&=5-3&&]]>\text{<acroref type="theorem" acro="CRN" />}\\
+<![CDATA[&=5-\rank{A}&&]]><acroref type="theorem" acro="RPNC" />\\
+<![CDATA[&=5-3&&]]><acroref type="theorem" acro="CRN" />\\
 <![CDATA[&=2]]>
 </alignmath>
 We could also use <acroref type="theorem" acro="BNS" /> and create a basis for $\nsp{A}$ with $n-r=5-3=2$ vectors (because the solutions are described with 2 free variables) and arrive at the dimension as the size of this basis.
 <solution contributor="robertbeezer">(1) We will use the three criteria of <acroref type="theorem" acro="TSS" />.  The zero vector of $M_{22}$ is the zero matrix, $\zeromatrix$ (<acroref type="definition" acro="ZM" />), which is a symmetric matrix.  So $S_{22}$ is not empty, since $\zeromatrix\in S_{22}$.<br /><br />
 Suppose that $A$ and $B$ are two matrices in $S_{22}$.  Then we know that $\transpose{A}=A$ and $\transpose{B}=B$.  We want to know if $A+B\in S_{22}$, so test $A+B$ for membership,
 <alignmath>
-<![CDATA[\transpose{\left(A+B\right)}&=\transpose{A}+\transpose{B}&&]]>\text{<acroref type="theorem" acro="TMA" />}\\
+<![CDATA[\transpose{\left(A+B\right)}&=\transpose{A}+\transpose{B}&&]]><acroref type="theorem" acro="TMA" />\\
 <![CDATA[&=A+B&&A,\,B\in S_{22}]]>
 </alignmath>
 So $A+B$ is symmetric and qualifies for membership in $S_{22}$.<br /><br />
 Suppose that $A\in S_{22}$ and $\alpha\in\complex{\null}$.  Is $\alpha A\in S_{22}$?  We know that $\transpose{A}=A$.  Now check that,
 <alignmath>
-<![CDATA[\transpose{\alpha A}&=\alpha\transpose{A}&&]]>\text{<acroref type="theorem" acro="TMSM" />}\\
+<![CDATA[\transpose{\alpha A}&=\alpha\transpose{A}&&]]><acroref type="theorem" acro="TMSM" />\\
 <![CDATA[&=\alpha A&&A\in S_{22}]]>
 </alignmath>
 So $\alpha A$ is also symmetric and qualifies for membership in $S_{22}$.<br /><br />

File src/section-DM.xml

 <alignmath>
 \matrixentry{\elemswap{i}{j}A}{k\ell}
 <![CDATA[&=\sum_{p=1}^{n}\matrixentry{\elemswap{i}{j}}{kp}\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\matrixentry{\elemswap{i}{j}}{kk}\matrixentry{A}{k\ell}+]]>
 \sum_{\substack{p=1\\p\neq k}}^{n}\matrixentry{\elemswap{i}{j}}{kp}\matrixentry{A}{p\ell}
-<![CDATA[&&]]>\text{<acroref type="property" acro="CACN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CACN" />\\
 <![CDATA[&=1\matrixentry{A}{k\ell}+]]>
 \sum_{\substack{p=1\\p\neq k}}^{n}0\matrixentry{A}{p\ell}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ELEM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ELEM" />\\
 <![CDATA[&=\matrixentry{A}{k\ell}]]>
 </alignmath>
 </p>
 <alignmath>
 \matrixentry{\elemswap{i}{j}A}{i\ell}
 <![CDATA[&=\sum_{p=1}^{n}\matrixentry{\elemswap{i}{j}}{ip}\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\matrixentry{\elemswap{i}{j}}{ij}\matrixentry{A}{j\ell}+]]>
 \sum_{\substack{p=1\\p\neq j}}^{n}\matrixentry{\elemswap{i}{j}}{ip}\matrixentry{A}{p\ell}
-<![CDATA[&&]]>\text{<acroref type="property" acro="CACN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CACN" />\\
 <![CDATA[&=1\matrixentry{A}{j\ell}+]]>
 \sum_{\substack{p=1\\p\neq j}}^{n}0\matrixentry{A}{p\ell}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ELEM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ELEM" />\\
 <![CDATA[&=\matrixentry{A}{j\ell}]]>
 </alignmath>
 </p>
 <alignmath>
 \matrixentry{\elemswap{i}{j}A}{j\ell}
 <![CDATA[&=\sum_{p=1}^{n}\matrixentry{\elemswap{i}{j}}{jp}\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\matrixentry{\elemswap{i}{j}}{ji}\matrixentry{A}{i\ell}+]]>
 \sum_{\substack{p=1\\p\neq i}}^{n}\matrixentry{\elemswap{i}{j}}{jp}\matrixentry{A}{p\ell}
-<![CDATA[&&]]>\text{<acroref type="property" acro="CACN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CACN" />\\
 <![CDATA[&=1\matrixentry{A}{i\ell}+\sum_{\substack{p=1\\p\neq i}}^{n}0\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ELEM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ELEM" />\\
 <![CDATA[&=\matrixentry{A}{i\ell}]]>
 </alignmath>
 </p>
 <alignmath>
 \matrixentry{\elemmult{\alpha}{i}A}{k\ell}
 <![CDATA[&=\sum_{p=1}^{n}\matrixentry{\elemmult{\alpha}{i}}{kp}\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\matrixentry{\elemmult{\alpha}{i}}{kk}\matrixentry{A}{k\ell}+]]>
 \sum_{\substack{p=1\\p\neq k}}^{n}\matrixentry{\elemmult{\alpha}{i}}{kp}\matrixentry{A}{p\ell}
-<![CDATA[&&]]>\text{<acroref type="property" acro="CACN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CACN" />\\
 <![CDATA[&=1\matrixentry{A}{k\ell}+\sum_{\substack{p=1\\p\neq k}}^{n}0\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ELEM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ELEM" />\\
 <![CDATA[&=\matrixentry{A}{k\ell}]]>
 </alignmath>
 </p>
 <alignmath>
 \matrixentry{\elemmult{\alpha}{i}A}{i\ell}
 <![CDATA[&=\sum_{p=1}^{n}\matrixentry{\elemmult{\alpha}{i}}{ip}\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\matrixentry{\elemmult{\alpha}{i}}{ii}\matrixentry{A}{i\ell}+]]>
 \sum_{\substack{p=1\\p\neq i}}^{n}\matrixentry{\elemmult{\alpha}{i}}{ip}\matrixentry{A}{p\ell}
-<![CDATA[&&]]>\text{<acroref type="property" acro="CACN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CACN" />\\
 <![CDATA[&=\alpha\matrixentry{A}{i\ell}+\sum_{\substack{p=1\\p\neq i}}^{n}0\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ELEM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ELEM" />\\
 <![CDATA[&=\alpha\matrixentry{A}{i\ell}]]>
 </alignmath>
 </p>
 <alignmath>
 \matrixentry{\elemadd{\alpha}{i}{j}A}{k\ell}
 <![CDATA[&=\sum_{p=1}^{n}\matrixentry{\elemadd{\alpha}{i}{j}}{kp}\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\matrixentry{\elemadd{\alpha}{i}{j}}{kk}\matrixentry{A}{k\ell}+]]>
 \sum_{\substack{p=1\\p\neq k}}^{n}\matrixentry{\elemadd{\alpha}{i}{j}}{kp}\matrixentry{A}{p\ell}
-<![CDATA[&&]]>\text{<acroref type="property" acro="CACN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CACN" />\\
 <![CDATA[&=1\matrixentry{A}{k\ell}+\sum_{\substack{p=1\\p\neq k}}^{n}0\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ELEM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ELEM" />\\
 <![CDATA[&=\matrixentry{A}{k\ell}]]>
 </alignmath>
 </p>
 <alignmath>
 \matrixentry{\elemadd{\alpha}{i}{j}A}{j\ell}
 <![CDATA[&=\sum_{p=1}^{n}\matrixentry{\elemadd{\alpha}{i}{j}}{jp}\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\matrixentry{\elemadd{\alpha}{i}{j}}{jj}\matrixentry{A}{j\ell}+\\]]>
 <![CDATA[&\quad\quad\matrixentry{\elemadd{\alpha}{i}{j}}{ji}\matrixentry{A}{i\ell}+]]>
 \sum_{\substack{p=1\\p\neq j,i}}^{n}\matrixentry{\elemadd{\alpha}{i}{j}}{jp}\matrixentry{A}{p\ell}
-<![CDATA[&&]]>\text{<acroref type="property" acro="CACN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CACN" />\\
 <![CDATA[&=1\matrixentry{A}{j\ell}+\alpha\matrixentry{A}{i\ell}+\sum_{\substack{p=1\\p\neq j,i}}^{n}0\matrixentry{A}{p\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ELEM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ELEM" />\\
 <![CDATA[&=\matrixentry{A}{j\ell}+\alpha\matrixentry{A}{i\ell}]]>
 </alignmath>
 </p>
 <![CDATA[(-1)^{2+1}\matrixentry{A}{21}&\detname{\submatrix{A}{2}{1}}+]]>
 (-1)^{2+2}\matrixentry{A}{22}\detname{\submatrix{A}{2}{2}}\\
 <![CDATA[&=-\matrixentry{A}{21}\matrixentry{A}{12}+\matrixentry{A}{22}\matrixentry{A}{11}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="DM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="DM" />\\
 <![CDATA[&=\matrixentry{A}{11}\matrixentry{A}{22}-\matrixentry{A}{12}\matrixentry{A}{21}\\]]>
 <![CDATA[&=]]>
-<![CDATA[\detname{A}&&]]>\text{<acroref type="theorem" acro="DMST" />}\\
+<![CDATA[\detname{A}&&]]><acroref type="theorem" acro="DMST" />\\
 </alignmath>
 </p>
 
 <![CDATA[&\detname{A}\\]]>
 <![CDATA[&\quad=]]>
 \sum_{j=1}^{n}(-1)^{1+j}\matrixentry{A}{1j}\detname{\submatrix{A}{1}{j}}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="DM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="DM" />\\
 <![CDATA[&\quad=]]>
 \sum_{j=1}^{n}(-1)^{1+j}\matrixentry{A}{1j}
 \sum_{\substack{1\leq\ell\leq n\\\ell\neq j}}
 \sum_{j=1}^{n}\sum_{\substack{1\leq\ell\leq n\\\ell\neq j}}
 (-1)^{j+i+\ell-\epsilon_{\ell j}}
 \matrixentry{A}{1j}\matrixentry{A}{i\ell}\detname{\submatrix{A}{1,i}{j,\ell}}
-<![CDATA[&&]]>\text{<acroref type="property" acro="DCN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="DCN" />\\
 <![CDATA[&\quad=]]>
 \sum_{\ell=1}^{n}\sum_{\substack{1\leq j\leq n\\j\neq\ell}}
 (-1)^{j+i+\ell-\epsilon_{\ell j}}
 \matrixentry{A}{1j}\matrixentry{A}{i\ell}\detname{\submatrix{A}{1,i}{j,\ell}}
-<![CDATA[&&]]>\text{<acroref type="property" acro="CACN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CACN" />\\
 <![CDATA[&\quad=]]>
 \sum_{\ell=1}^{n}(-1)^{i+\ell}\matrixentry{A}{i\ell}
 \sum_{\substack{1\leq j\leq n\\j\neq\ell}}
 (-1)^{j-\epsilon_{\ell j}}
 \matrixentry{A}{1j}\detname{\submatrix{A}{1,i}{j,\ell}}
-<![CDATA[&&]]>\text{<acroref type="property" acro="DCN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="DCN" />\\
 <![CDATA[&\quad=]]>
 \sum_{\ell=1}^{n}(-1)^{i+\ell}\matrixentry{A}{i\ell}
 \sum_{\substack{1\leq j\leq n\\j\neq\ell}}
 <![CDATA[&&\text{$2\epsilon_{\ell j}$ is even}\\]]>
 <![CDATA[&\quad=]]>
 \sum_{\ell=1}^{n}(-1)^{i+\ell}\matrixentry{A}{i\ell}\detname{\submatrix{A}{i}{\ell}}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="DM" />}
+<![CDATA[&&]]><acroref type="definition" acro="DM" />
 </alignmath>
 </p>
 
 <![CDATA[&=]]>
 \frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}(-1)^{i+j}
 \matrixentry{\transpose{A}}{ij}\detname{\submatrix{\transpose{A}}{i}{j}}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="DER" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="DER" />\\
 <![CDATA[&=]]>
 \frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}(-1)^{i+j}
 \matrixentry{A}{ji}\detname{\submatrix{\transpose{A}}{i}{j}}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="TM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="TM" />\\
 <![CDATA[&=]]>
 \frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}(-1)^{i+j}
 \matrixentry{A}{ji}\detname{\transpose{\left(\submatrix{A}{j}{i}\right)}}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="TM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="TM" />\\
 <![CDATA[&=]]>
 \frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}(-1)^{i+j}
 \matrixentry{A}{ji}\detname{\submatrix{A}{j}{i}}
 <![CDATA[&=]]>
 \frac{1}{n}\sum_{j=1}^{n}\sum_{i=1}^{n}(-1)^{j+i}
 \matrixentry{A}{ji}\detname{\submatrix{A}{j}{i}}
-<![CDATA[&&]]>\text{<acroref type="property" acro="CACN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CACN" />\\
 <![CDATA[&=]]>
 \frac{1}{n}\sum_{j=1}^{n}\detname{A}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="DER" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="DER" />\\
 <![CDATA[&=\detname{A}]]>
 </alignmath>
 </p>
 <alignmath>
 \detname{A}
 <![CDATA[&=]]>
-<![CDATA[\detname{\transpose{A}}&&]]>\text{<acroref type="theorem" acro="DT" />}\\
+<![CDATA[\detname{\transpose{A}}&&]]><acroref type="theorem" acro="DT" />\\
 <![CDATA[&=]]>
 \sum_{i=1}^{n}(-1)^{j+i}\matrixentry{\transpose{A}}{ji}\detname{\submatrix{\transpose{A}}{j}{i}}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="DER" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="DER" />\\
 <![CDATA[&=]]>
 \sum_{i=1}^{n}(-1)^{j+i}\matrixentry{\transpose{A}}{ji}\detname{\transpose{\left(\submatrix{A}{i}{j}\right)}}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="TM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="TM" />\\
 <![CDATA[&=]]>
 \sum_{i=1}^{n}(-1)^{j+i}\matrixentry{\transpose{A}}{ji}\detname{\submatrix{A}{i}{j}}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="DT" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="DT" />\\
 <![CDATA[&=]]>
 \sum_{i=1}^{n}(-1)^{i+j}\matrixentry{A}{ij}\detname{\submatrix{A}{i}{j}}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="TM" />}
+<![CDATA[&&]]><acroref type="definition" acro="TM" />
 </alignmath>
 </p>
 

File src/section-EE.xml

 <alignmath>
 A\vect{u}
 <![CDATA[&=A(30\vect{x})\\]]>
-<![CDATA[&=30A\vect{x}&&]]>\text{<acroref type="theorem" acro="MMSMM" />}\\
+<![CDATA[&=30A\vect{x}&&]]><acroref type="theorem" acro="MMSMM" />\\
 <![CDATA[&=30(4\vect{x})&&\text{$\vect{x}$ an eigenvector of $A$}\\]]>
-<![CDATA[&=4(30\vect{x})&&]]>\text{<acroref type="property" acro="SMAM" />}\\
+<![CDATA[&=4(30\vect{x})&&]]><acroref type="property" acro="SMAM" />\\
 <![CDATA[&=4\vect{u}]]>
 </alignmath>
 so that $\vect{u}$ is also an eigenvector of $A$ for the same eigenvalue, $\lambda=4$.</p>
 <alignmath>
 A\vect{v}
 <![CDATA[&=A(\vect{z}+\vect{w})\\]]>
-<![CDATA[&=A\vect{z}+A\vect{w}&&]]>\text{<acroref type="theorem" acro="MMDAA" />}\\
+<![CDATA[&=A\vect{z}+A\vect{w}&&]]><acroref type="theorem" acro="MMDAA" />\\
 <![CDATA[&=2\vect{z}+2\vect{w}&&\text{$\vect{z}$, $\vect{w}$ eigenvectors of $A$}\\]]>
-<![CDATA[&=2(\vect{z}+\vect{w})&&]]>\text{<acroref type="property" acro="DVAC" />}\\
+<![CDATA[&=2(\vect{z}+\vect{w})&&]]><acroref type="property" acro="DVAC" />\\
 <![CDATA[&=2\vect{v}]]>
 </alignmath>
 so that $\vect{v}$ is also an eigenvector of $A$ for the eigenvalue $\lambda=2$.  So it would appear that the set of eigenvectors that are associated with a fixed eigenvalue is closed under the vector space operations of $\complex{n}$.  Hmmm.</p>
 <alignmath>
 <![CDATA[\zerovector&=a_0\vect{x}+a_1A\vect{x}+a_2A^2\vect{x}+\cdots+a_nA^n\vect{x}\\]]>
 <![CDATA[&=a_0\vect{x}+a_1A\vect{x}+a_2A^2\vect{x}+\cdots+a_mA^m\vect{x}&&\text{$a_i=0$ for $i>m$}\\]]>
-<![CDATA[&=\left(a_0I_n+a_1A+a_2A^2+\cdots+a_mA^m\right)\vect{x}&&]]>\text{<acroref type="theorem" acro="MMDAA" />}\\
+<![CDATA[&=\left(a_0I_n+a_1A+a_2A^2+\cdots+a_mA^m\right)\vect{x}&&]]><acroref type="theorem" acro="MMDAA" />\\
 <![CDATA[&=p(A)\vect{x}&&\text{Definition of $p(x)$}\\]]>
 <![CDATA[&=(A-b_mI_n)(A-b_{m-1}I_n)\cdots(A-b_2I_n)(A-b_1I_n)\vect{x}]]>
 </alignmath>
 which allows us to write
 <alignmath>
 A\vect{z}
-<![CDATA[&=(A+\zeromatrix)\vect{z}&&]]>\text{<acroref type="property" acro="ZM" />}\\
-<![CDATA[&=(A-b_kI_n+b_kI_n)\vect{z}&&]]>\text{<acroref type="property" acro="AIM" />}\\
-<![CDATA[&=(A-b_kI_n)\vect{z}+b_kI_n\vect{z}&&]]>\text{<acroref type="theorem" acro="MMDAA" />}\\
+<![CDATA[&=(A+\zeromatrix)\vect{z}&&]]><acroref type="property" acro="ZM" />\\
+<![CDATA[&=(A-b_kI_n+b_kI_n)\vect{z}&&]]><acroref type="property" acro="AIM" />\\
+<![CDATA[&=(A-b_kI_n)\vect{z}+b_kI_n\vect{z}&&]]><acroref type="theorem" acro="MMDAA" />\\
 <![CDATA[&=\zerovector+b_kI_n\vect{z}&&\text{Defining property of $\vect{z}$}\\]]>
-<![CDATA[&=b_kI_n\vect{z}&&]]>\text{<acroref type="property" acro="ZM" />}\\
-<![CDATA[&=b_k\vect{z}&&]]>\text{<acroref type="theorem" acro="MMIM" />}
+<![CDATA[&=b_kI_n\vect{z}&&]]><acroref type="property" acro="ZM" />\\
+<![CDATA[&=b_k\vect{z}&&]]><acroref type="theorem" acro="MMIM" />
 </alignmath>
 </p>
 
 <![CDATA[-13-x & -8 & -4\\]]>
 <![CDATA[12 & 7-x & 4\\]]>
 <![CDATA[24 & 16 & 7-x]]>
-<![CDATA[\end{vmatrix}&&]]>\text{<acroref type="definition" acro="CP" />}\\
+<![CDATA[\end{vmatrix}&&]]><acroref type="definition" acro="CP" />\\
 <![CDATA[&=]]>
 (-13-x)
 \begin{vmatrix}
 \begin{vmatrix}
 <![CDATA[12  & 4\\]]>
 <![CDATA[24  & 7-x]]>
-<![CDATA[\end{vmatrix}&&]]>\text{<acroref type="definition" acro="DM" />}\\
+<![CDATA[\end{vmatrix}&&]]><acroref type="definition" acro="DM" />\\
 <![CDATA[&\quad\quad]]>
 +(-4)
 \begin{vmatrix}
 <![CDATA[12 & 7-x\\]]>
 <![CDATA[24 & 16]]>
 \end{vmatrix}\\
-<![CDATA[&=(-13-x)((7-x)(7-x)-4(16))&&]]>\text{<acroref type="theorem" acro="DMST" />}\\
+<![CDATA[&=(-13-x)((7-x)(7-x)-4(16))&&]]><acroref type="theorem" acro="DMST" />\\
 <![CDATA[&\quad\quad +(-8)(-1)(12(7-x)-4(24))\\]]>
 <![CDATA[&\quad\quad +(-4)(12(16)-(7-x)(24))\\]]>
 <![CDATA[&=3+5x+x^2-x^3\\]]>
 <alignmath>
 <![CDATA[&\text{$\lambda$ is an eigenvalue of $A$}\\]]>
 <![CDATA[&\iff\text{ there exists $\vect{x}\neq\zerovector$ so that $A\vect{x}=\lambda\vect{x}$}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="EEM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="EEM" />\\
 <![CDATA[&\iff \text{ there exists $\vect{x}\neq\zerovector$ so that $A\vect{x}-\lambda\vect{x}=\zerovector$}\\]]>
 <![CDATA[&\iff \text{ there exists $\vect{x}\neq\zerovector$ so that $A\vect{x}-\lambda I_n\vect{x}=\zerovector$}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMIM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MMIM" />\\
 <![CDATA[&\iff\text{ there exists $\vect{x}\neq\zerovector$ so that $(A-\lambda I_n)\vect{x}=\zerovector$}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMDAA" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MMDAA" />\\
 <![CDATA[&\iff A-\lambda I_n\text{ is singular}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="NM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="NM" />\\
 <![CDATA[&\iff\detname{A-\lambda I_n}=0]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="SMZD" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="SMZD" />\\
 <![CDATA[&\iff\charpoly{A}{\lambda}=0]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="CP" />}
+<![CDATA[&&]]><acroref type="definition" acro="CP" />
 </alignmath>
 </p>
 
 
 <p>Suppose that $\vect{x},\,\vect{y}\in\eigenspace{A}{\lambda}$, that is, $\vect{x}$ and $\vect{y}$ are two eigenvectors of $A$ for $\lambda$.  Then
 <alignmath>
-<![CDATA[A\left(\vect{x}+\vect{y}\right)&=A\vect{x}+A\vect{y}&&]]>\text{<acroref type="theorem" acro="MMDAA" />}\\
+<![CDATA[A\left(\vect{x}+\vect{y}\right)&=A\vect{x}+A\vect{y}&&]]><acroref type="theorem" acro="MMDAA" />\\
 <![CDATA[&=\lambda\vect{x}+\lambda\vect{y}&&\text{$\vect{x},\,\vect{y}$ eigenvectors of $A$}\\]]>
-<![CDATA[&=\lambda\left(\vect{x}+\vect{y}\right)&&]]>\text{<acroref type="property" acro="DVAC" />}
+<![CDATA[&=\lambda\left(\vect{x}+\vect{y}\right)&&]]><acroref type="property" acro="DVAC" />
 </alignmath>
 </p>
 
 
 <p>Suppose that $\alpha\in\complex{\null}$, and that $\vect{x}\in\eigenspace{A}{\lambda}$, that is, $\vect{x}$ is an eigenvector of $A$ for $\lambda$.  Then
 <alignmath>
-<![CDATA[A\left(\alpha\vect{x}\right)&=\alpha\left(A\vect{x}\right)&&]]>\text{<acroref type="theorem" acro="MMSMM" />}\\
+<![CDATA[A\left(\alpha\vect{x}\right)&=\alpha\left(A\vect{x}\right)&&]]><acroref type="theorem" acro="MMSMM" />\\
 <![CDATA[&=\alpha\lambda\vect{x}&&\text{$\vect{x}$ an eigenvector of $A$}\\]]>
-<![CDATA[&=\lambda\left(\alpha\vect{x}\right)&&]]>\text{<acroref type="property" acro="SMAC" />}
+<![CDATA[&=\lambda\left(\alpha\vect{x}\right)&&]]><acroref type="property" acro="SMAC" />
 </alignmath></p>
 
 <p>So either $\alpha\vect{x}=\zerovector$, or $\alpha\vect{x}$ is an eigenvector of $A$ for $\lambda$ (<acroref type="definition" acro="EEM" />).  So, in either event,  $\alpha\vect{x}\in\eigenspace{A}{\lambda}$, and we have scalar closure.</p>
 <proof>
 <p>The conclusion of this theorem is an equality of sets, so normally we would follow the advice of <acroref type="definition" acro="SE" />.  However, in this case we can construct a sequence of equivalences which will together provide the two subset inclusions we need.  First, notice that $\zerovector\in\eigenspace{A}{\lambda}$ by <acroref type="definition" acro="EM" /> and $\zerovector\in\nsp{A-\lambda I_n}$ by <acroref type="theorem" acro="HSC" />.  Now consider any nonzero vector $\vect{x}\in\complex{n}$,
 <alignmath>
-<![CDATA[\vect{x}\in\eigenspace{A}{\lambda}&\iff A\vect{x}=\lambda\vect{x}&&]]>\text{<acroref type="definition" acro="EM" />}\\
+<![CDATA[\vect{x}\in\eigenspace{A}{\lambda}&\iff A\vect{x}=\lambda\vect{x}&&]]><acroref type="definition" acro="EM" />\\
 <![CDATA[&\iff A\vect{x}-\lambda\vect{x}=\zerovector\\]]>
-<![CDATA[&\iff A\vect{x}-\lambda I_n\vect{x}=\zerovector&&]]>\text{<acroref type="theorem" acro="MMIM" />}\\
-<![CDATA[&\iff\left(A-\lambda I_n\right)\vect{x}=\zerovector&&]]>\text{<acroref type="theorem" acro="MMDAA" />}\\
-<![CDATA[&\iff\vect{x}\in\nsp{A-\lambda I_n}&&]]>\text{<acroref type="definition" acro="NSM" />}
+<![CDATA[&\iff A\vect{x}-\lambda I_n\vect{x}=\zerovector&&]]><acroref type="theorem" acro="MMIM" />\\
+<![CDATA[&\iff\left(A-\lambda I_n\right)\vect{x}=\zerovector&&]]><acroref type="theorem" acro="MMDAA" />\\
+<![CDATA[&\iff\vect{x}\in\nsp{A-\lambda I_n}&&]]><acroref type="definition" acro="NSM" />
 </alignmath>
 </p>
 
 <solution contributor="robertbeezer">First compute the characteristic polynomial,
 <alignmath>
 \charpoly{C}{x}
-<![CDATA[&=\detname{C-xI_2}&&]]>\text{<acroref type="definition" acro="CP" />}\\
+<![CDATA[&=\detname{C-xI_2}&&]]><acroref type="definition" acro="CP" />\\
 <![CDATA[&=]]>
 \begin{vmatrix}
 <![CDATA[ -1-x & 2 \\]]>
 <![CDATA[ -6    & 6-x]]>
 \end{vmatrix}\\
-<![CDATA[&=(-1-x)(6-x)-(2)(-6)&&]]>\text{<acroref type="theorem" acro="DMST" />}\\
+<![CDATA[&=(-1-x)(6-x)-(2)(-6)&&]]><acroref type="theorem" acro="DMST" />\\
 <![CDATA[&=x^2-5x+6\\]]>
 <![CDATA[&=(x-3)(x-2)]]>
 </alignmath>
 </problem>
 <solution contributor="robertbeezer">The characteristic polynomial of $B$ is
 <alignmath>
-<![CDATA[\charpoly{B}{x}&=\detname{B-xI_2}&&]]>\text{<acroref type="definition" acro="CP" />}\\
+<![CDATA[\charpoly{B}{x}&=\detname{B-xI_2}&&]]><acroref type="definition" acro="CP" />\\
 <![CDATA[&=]]>
 \begin{vmatrix}
 <![CDATA[-12-x&30\\-5&13-x]]>
 \end{vmatrix}\\
-<![CDATA[&=(-12-x)(13-x)-(30)(-5)&&]]>\text{<acroref type="theorem" acro="DMST" />}\\
+<![CDATA[&=(-12-x)(13-x)-(30)(-5)&&]]><acroref type="theorem" acro="DMST" />\\
 <![CDATA[&=x^2-x-6\\]]>
 <![CDATA[&=(x-3)(x+2)]]>
 </alignmath>
 <![CDATA[2-x & -1\\]]>
 <![CDATA[1 & 1-x]]>
 \end{vmatrix}\\
-<![CDATA[&=(2-x)(1-x)-(1)(-1)&&]]>\text{<acroref type="theorem" acro="DMST" />}\\
+<![CDATA[&=(2-x)(1-x)-(1)(-1)&&]]><acroref type="theorem" acro="DMST" />\\
 <![CDATA[&=x^2-3x+3\\]]>
 <![CDATA[&=\left(x-\frac{3+\sqrt{3}i}{2}\right)\left(x-\frac{3-\sqrt{3}i}{2}\right)]]>
 </alignmath>
 <![CDATA[&=A^2\vect{x}&&\text{$A$ is idempotent}\\]]>
 <![CDATA[&=A(A\vect{x})\\]]>
 <![CDATA[&=A(\lambda\vect{x})&&\text{$\vect{x}$ eigenvector of $A$}\\]]>
-<![CDATA[&=\lambda(A\vect{x})&&]]>\text{<acroref type="theorem" acro="MMSMM" />}\\
+<![CDATA[&=\lambda(A\vect{x})&&]]><acroref type="theorem" acro="MMSMM" />\\
 <![CDATA[&=\lambda(\lambda\vect{x})&&\text{$\vect{x}$ eigenvector of $A$}\\]]>
 <![CDATA[&=\lambda^2\vect{x}]]>
 <intertext>From this we get</intertext>
 <![CDATA[\zerovector&=\lambda^2\vect{x}-\lambda\vect{x}\\]]>
-<![CDATA[&=(\lambda^2-\lambda)\vect{x}&&]]>\text{<acroref type="property" acro="DSAC" />}\\
+<![CDATA[&=(\lambda^2-\lambda)\vect{x}&&]]><acroref type="property" acro="DSAC" />\\
 </alignmath>
 Since $\vect{x}$ is an eigenvector, it is nonzero, and <acroref type="theorem" acro="SMEZV" /> leaves us with the conclusion that $\lambda^2-\lambda=0$, and the solutions to this quadratic polynomial equation in $\lambda$ are $\lambda=0$ and $\lambda=1$.<br /><br />
 The matrix
 <solution contributor="robertbeezer">Note in the following that the scalar multiple of a matrix is equivalent to multiplying each of the rows by that scalar, so we actually apply <acroref type="theorem" acro="DRCM" /> multiple times below (and are passing up an opportunity to do a proof by induction in the process, which maybe you'd like to do yourself?).
 <alignmath>
 \charpoly{A}{x}
-<![CDATA[&=\detname{A-xI_n}&&]]>\text{<acroref type="definition" acro="CP" />}\\
-<![CDATA[&=\detname{(-1)(xI_n-A)}&&]]>\text{<acroref type="definition" acro="MSM" />}\\
-<![CDATA[&=(-1)^{n}\detname{xI_n-A}&&]]>\text{<acroref type="theorem" acro="DRCM" />}\\
+<![CDATA[&=\detname{A-xI_n}&&]]><acroref type="definition" acro="CP" />\\
+<![CDATA[&=\detname{(-1)(xI_n-A)}&&]]><acroref type="definition" acro="MSM" />\\
+<![CDATA[&=(-1)^{n}\detname{xI_n-A}&&]]><acroref type="theorem" acro="DRCM" />\\
 <![CDATA[&=(-1)^{n}r_A(x)]]>
 </alignmath>
 Since the polynomials are scalar multiples of each other, their roots will be identical, so either polynomial could be used in <acroref type="theorem" acro="EMRCP" />.<br /><br />
 <alignmath>
 \vect{x}
 <![CDATA[&=1\vect{x}]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="O" />}\\
+<![CDATA[&&]]><acroref type="property" acro="O" />\\
 <![CDATA[&=\frac{1}{\lambda-\rho}\left(\lambda-\rho\right)\vect{x}]]>
 <![CDATA[&&\lambda\neq\rho,\ \lambda-\rho\neq 0\\]]>
 <![CDATA[&=\frac{1}{\lambda-\rho}\left(\lambda\vect{x}-\rho\vect{x}\right)]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="DSAC" />}\\
+<![CDATA[&&]]><acroref type="property" acro="DSAC" />\\
 <![CDATA[&=\frac{1}{\lambda-\rho}\left(A\vect{x}-A\vect{x}\right)]]>
 <![CDATA[&&\text{$\vect{x}$ eigenvector of $A$ for $\lambda$, $\rho$}\\]]>
 <![CDATA[&=\frac{1}{\lambda-\rho}\left(\zerovector\right)\\]]>
 <![CDATA[&=\zerovector]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="ZVSM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="ZVSM" />\\
 </alignmath>
 So $\vect{x}=\zerovector$, and trivially, $\vect{x}\in\set{\zerovector}$.
 </solution>

File src/section-FS.xml

 <alignmath>
 \transpose{\zerovector}
 <![CDATA[&=\transpose{\left(\transpose{A}\vect{y}\right)}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="LNS" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="LNS" />\\
 <![CDATA[&=\transpose{\vect{y}}\transpose{\left(\transpose{A}\right)}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMT" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MMT" />\\
 <![CDATA[&=\transpose{\vect{y}}A]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="TT" />}
+<![CDATA[&&]]><acroref type="theorem" acro="TT" />
 </alignmath>
 </p>
 
 <![CDATA[&=\vectorentry{J\vect{y}}{r+k}]]>
 <![CDATA[&&\text{$L$ a submatrix of $J$}\\]]>
 <![CDATA[&=\vectorentry{B\vect{x}}{r+k}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="PEEF" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="PEEF" />\\
 <![CDATA[&=\vectorentry{\zeromatrix\vect{x}}{k}]]>
 <![CDATA[&&\text{Zero matrix a submatrix of $B$}\\]]>
 <![CDATA[&=\vectorentry{\zerovector}{k}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMZM" />}
+<![CDATA[&&]]><acroref type="theorem" acro="MMZM" />
 </alignmath>
 </p>
 
 <![CDATA[&=\vectorentry{\zeromatrix\vect{x}}{k-r}]]>
 <![CDATA[&&\text{Zero matrix a submatrix of $B$}\\]]>
 <![CDATA[&=\vectorentry{\zerovector}{k-r}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMZM" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MMZM" />\\
 <![CDATA[&=\vectorentry{L\vect{y}}{k-r}]]>
 <![CDATA[&&\text{$\vect{y}$ in $\nsp{L}$}\\]]>
 <![CDATA[&=\vectorentry{J\vect{y}}{k}]]>
 <alignmath>
 \vectorentry{\transpose{A}\vect{y}}{i}
 <![CDATA[&=\sum_{k=1}^{m}\matrixentry{\transpose{A}}{ik}\vectorentry{\vect{y}}{k}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\sum_{k=1}^{m}\matrixentry{\transpose{A}}{ik}\vectorentry{\transpose{L}\vect{w}}{k}]]>
 <![CDATA[&&\text{Definition of $\vect{w}$}\\]]>
 <![CDATA[&=\sum_{k=1}^{m}\matrixentry{\transpose{A}}{ik}\sum_{\ell=1}^{m-r}\matrixentry{\transpose{L}}{k\ell}\vectorentry{\vect{w}}{\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\sum_{k=1}^{m}\sum_{\ell=1}^{m-r}\matrixentry{\transpose{A}}{ik}\matrixentry{\transpose{L}}{k\ell}\vectorentry{\vect{w}}{\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="DCN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="DCN" />\\
 <![CDATA[&=\sum_{\ell=1}^{m-r}\sum_{k=1}^{m}\matrixentry{\transpose{A}}{ik}\matrixentry{\transpose{L}}{k\ell}\vectorentry{\vect{w}}{\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="CACN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CACN" />\\
 <![CDATA[&=\sum_{\ell=1}^{m-r}\left(\sum_{k=1}^{m}\matrixentry{\transpose{A}}{ik}\matrixentry{\transpose{L}}{k\ell}\right)\vectorentry{\vect{w}}{\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="DCN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="DCN" />\\
 <![CDATA[&=\sum_{\ell=1}^{m-r}\left(\sum_{k=1}^{m}\matrixentry{\transpose{A}}{ik}\matrixentry{\transpose{J}}{k,r+\ell}\right)\vectorentry{\vect{w}}{\ell}]]>
 <![CDATA[&&\text{$L$ a submatrix of $J$}\\]]>
 <![CDATA[&=\sum_{\ell=1}^{m-r}\matrixentry{\transpose{A}\transpose{J}}{i,r+\ell}\vectorentry{\vect{w}}{\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\sum_{\ell=1}^{m-r}\matrixentry{\transpose{\left(JA\right)}}{i,r+\ell}\vectorentry{\vect{w}}{\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMT" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MMT" />\\
 <![CDATA[&=\sum_{\ell=1}^{m-r}\matrixentry{\transpose{B}}{i,r+\ell}\vectorentry{\vect{w}}{\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="PEEF" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="PEEF" />\\
 <![CDATA[&=\sum_{\ell=1}^{m-r}0\vectorentry{\vect{w}}{\ell}]]>
 <![CDATA[&&\text{Zero rows in $B$}\\]]>
 <![CDATA[&=0]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="ZCN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="ZCN" />\\
 <![CDATA[&=\vectorentry{\zerovector}{i}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ZCV" />}
+<![CDATA[&&]]><acroref type="definition" acro="ZCV" />
 </alignmath>
 </p>
 
 <alignmath>
 \vectorentry{\transpose{C}\vect{z}}{j}
 <![CDATA[&=\sum_{k=1}^{r}\matrixentry{\transpose{C}}{jk}\vectorentry{\vect{z}}{k}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\sum_{k=1}^{r}\matrixentry{\transpose{C}}{jk}\vectorentry{\vect{z}}{k}+]]>
 \sum_{\ell=1}^{m-r}\matrixentry{\zeromatrix}{j\ell}\vectorentry{\vect{w}}{\ell}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ZM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="ZM" />\\
 <![CDATA[&=\sum_{k=1}^{r}\matrixentry{\transpose{B}}{jk}\vectorentry{\vect{z}}{k}+]]>
 \sum_{\ell=1}^{m-r}\matrixentry{\transpose{B}}{j,r+\ell}\vectorentry{\vect{w}}{\ell}
 <![CDATA[&&\text{$C$, $\zeromatrix$ submatrices of $B$}\\]]>
 <![CDATA[&=\sum_{k=1}^{m}\matrixentry{\transpose{B}}{jk}\vectorentry{\vect{x}}{k}]]>
 <![CDATA[&&\text{Combine sums}\\]]>
 <![CDATA[&=\sum_{k=1}^{m}\matrixentry{\transpose{\left(JA\right)}}{jk}\vectorentry{\vect{x}}{k}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="PEEF" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="PEEF" />\\
 <![CDATA[&=\sum_{k=1}^{m}\matrixentry{\transpose{A}\transpose{J}}{jk}\vectorentry{\vect{x}}{k}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="MMT" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="MMT" />\\
 <![CDATA[&=\sum_{k=1}^{m}\sum_{\ell=1}^{m}\matrixentry{\transpose{A}}{j\ell}\matrixentry{\transpose{J}}{\ell k}\vectorentry{\vect{x}}{k}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\sum_{\ell=1}^{m}\sum_{k=1}^{m}\matrixentry{\transpose{A}}{j\ell}\matrixentry{\transpose{J}}{\ell k}\vectorentry{\vect{x}}{k}]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="CACN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CACN" />\\
 <![CDATA[&=\sum_{\ell=1}^{m}\matrixentry{\transpose{A}}{j\ell}\left(\sum_{k=1}^{m}\matrixentry{\transpose{J}}{\ell k}\vectorentry{\vect{x}}{k}\right)]]>
-<![CDATA[&&]]>\text{<acroref type="property" acro="DCN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="DCN" />\\
 <![CDATA[&=\sum_{\ell=1}^{m}\matrixentry{\transpose{A}}{j\ell}\vectorentry{\transpose{J}\vect{x}}{\ell}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\sum_{\ell=1}^{m}\matrixentry{\transpose{A}}{j\ell}\vectorentry{\vect{y}}{\ell}]]>
 <![CDATA[&&\text{Definition of $\vect{x}$}\\]]>
 <![CDATA[&=\vectorentry{\transpose{A}\vect{y}}{j}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\vectorentry{\zerovector}{j}]]>
 <![CDATA[&&\text{$\vect{y}\in\lns{A}$}]]>
 </alignmath>
 <![CDATA[&=\vectorentry{\transpose{J}\vect{x}}{i}]]>
 <![CDATA[&&\text{Definition of $\vect{x}$}\\]]>
 <![CDATA[&=\sum_{k=1}^{m}\matrixentry{\transpose{J}}{ik}\vectorentry{\vect{x}}{k}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 <![CDATA[&=\sum_{k=1}^{r}\matrixentry{\transpose{J}}{ik}\vectorentry{\vect{x}}{k}+]]>
 \sum_{k=r+1}^{m}\matrixentry{\transpose{J}}{ik}\vectorentry{\vect{x}}{k}
 <![CDATA[&&\text{Break apart sum}\\]]>
 <![CDATA[&=0+\sum_{\ell=1}^{m-r}\matrixentry{\transpose{L}}{i,\ell}\vectorentry{\vect{w}}{\ell}]]>
 <![CDATA[&&\text{$L$ a submatrix of $J$}\\]]>
 <![CDATA[&=\vectorentry{\transpose{L}\vect{w}}{i}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="EMP" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="EMP" />\\
 </alignmath>
 </p>
 
 \colvector{-1\\6\\0\\1\\0\\0},\,
 \colvector{-3\\1\\0\\0\\-2\\1}
 }}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="BNS" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="BNS" />\\
 <![CDATA[\rsp{A}&=\rsp{C}=]]>
 \spn{\set{
 \colvector{1\\0\\2\\1\\0\\3 },\,
 \colvector{0\\1\\4\\-6\\0\\-1},\,
 \colvector{0\\0\\0\\0\\1\\2}
 }}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="BRS" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="BRS" />\\
 <![CDATA[\csp{A}&=\nsp{L}=]]>
 \spn{\set{
 \colvector{-2\\1\\0\\0},\,
 \colvector{-2\\0\\1\\0},\,
 \colvector{-1\\0\\0\\1}
 }}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="BNS" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="BNS" />\\
 <![CDATA[\lns{A}&=\rsp{L}=]]>
 \spn{\set{
 \colvector{1\\2\\2\\1}
 }}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="BRS" />}
+<![CDATA[&&]]><acroref type="theorem" acro="BRS" />
 </alignmath></p>
 
 <p>Boom!</p>
 \spn{\set{
 \colvector{-2\\3\\-1\\2\\1}
 }}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="BNS" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="BNS" />\\
 <![CDATA[\rsp{A}&=\rsp{C}=]]>
 \spn{\set{
 \colvector{1 \\ 0 \\ 0 \\ 0 \\ 2},\,
 \colvector{0 \\ 0 \\ 1 \\ 0 \\ 1},\,
 \colvector{0 \\ 0 \\ 0 \\ 1 \\ -2}
 }}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="BRS" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="BRS" />\\
 <![CDATA[\csp{A}&=\nsp{L}=]]>
 \spn{\set{
 \colvector{-3\\2\\1\\0\\0\\0},\,
 \colvector{-3\\-1\\0\\0\\1\\0},\,
 \colvector{-1\\1\\0\\0\\0\\1}
 }}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="BNS" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="BNS" />\\
 <![CDATA[\lns{A}&=\rsp{L}=]]>
 \spn{\set{
 \colvector{1 \\ 0 \\ 3 \\ -1 \\ 3 \\ 1},\,
 \colvector{0 \\ 1 \\ -2 \\ 1 \\ 1 \\ -1}
 }}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="BRS" />}
+<![CDATA[&&]]><acroref type="theorem" acro="BRS" />
 </alignmath>
 </p>
 
 <![CDATA[\nsp{G}=\nsp{C}&=]]>
 \spn{\emptyset}
 =\set{\zerovector}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="BNS" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="BNS" />\\
 <![CDATA[\rsp{G}=\rsp{C}&=]]>
 \spn{\set{
 \colvector{1 \\ 0},\,
 \colvector{0 \\ 1}
 }}
 =\complex{2}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="BRS" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="BRS" />\\
 <![CDATA[\csp{G}=\nsp{L}&=]]>
 \spn{\set{
 \colvector{0\\-1\\-1\\1\\0},\,
 \colvector{\frac{1}{3}\\\frac{1}{3}\\1\\0\\1}
-<![CDATA[}}&&]]>\text{<acroref type="theorem" acro="BNS" />}\\
+<![CDATA[}}&&]]><acroref type="theorem" acro="BNS" />\\
 <![CDATA[&=\spn{\set{]]>
 \colvector{0\\-1\\-1\\1\\0},\,
 \colvector{1\\1\\3\\0\\3}
 \colvector{1 \\ 0 \\ 0 \\ 0 \\ -\frac{1}{3}},\,
 \colvector{0 \\ 1  \\ 0 \\ 1 \\ -\frac{1}{3}},\,
 \colvector{0 \\ 0 \\ 1  \\ 1 \\ -1}
-<![CDATA[}}&&]]>\text{<acroref type="theorem" acro="BRS" />}\\
+<![CDATA[}}&&]]><acroref type="theorem" acro="BRS" />\\
 <![CDATA[&=\spn{\set{]]>
 \colvector{3 \\ 0 \\ 0 \\ 0 \\ -1},\,
 \colvector{0 \\ 3  \\ 0 \\ 3\ \\ -1},\,
 (a)
 <alignmath>
 \rsp{A}
-<![CDATA[&=\rsp{C}&&]]>\text{<acroref type="theorem" acro="FS" />}\\
-<![CDATA[&=\spn{\colvector{1\\0\\2},\,\colvector{0\\1\\3}}&&]]>\text{<acroref type="theorem" acro="BRS" />}
+<![CDATA[&=\rsp{C}&&]]><acroref type="theorem" acro="FS" />\\
+<![CDATA[&=\spn{\colvector{1\\0\\2},\,\colvector{0\\1\\3}}&&]]><acroref type="theorem" acro="BRS" />
 </alignmath>
 (b)
 <alignmath>
 \csp{A}
-<![CDATA[&=\nsp{L}&&]]>\text{<acroref type="theorem" acro="FS" />}\\
-<![CDATA[&=\spn{\colvector{1\\-1\\1\\0},\,\colvector{1\\-3\\0\\1}}&&]]>\text{<acroref type="theorem" acro="BNS" />}
+<![CDATA[&=\nsp{L}&&]]><acroref type="theorem" acro="FS" />\\
+<![CDATA[&=\spn{\colvector{1\\-1\\1\\0},\,\colvector{1\\-3\\0\\1}}&&]]><acroref type="theorem" acro="BNS" />
 </alignmath>
 (c)
 <alignmath>
 \nsp{A}
-<![CDATA[&=\nsp{C}&&]]>\text{<acroref type="theorem" acro="FS" />}\\
-<![CDATA[&=\spn{\colvector{-2\\-3\\1}}&&]]>\text{<acroref type="theorem" acro="BNS" />}
+<![CDATA[&=\nsp{C}&&]]><acroref type="theorem" acro="FS" />\\
+<![CDATA[&=\spn{\colvector{-2\\-3\\1}}&&]]><acroref type="theorem" acro="BNS" />
 </alignmath>
 (d)
 <alignmath>
 \lns{A}
-<![CDATA[&=\rsp{L}&&]]>\text{<acroref type="theorem" acro="FS" />}\\
-<![CDATA[&=\spn{\colvector{1\\0\\-1\\-1},\,\colvector{0\\1\\1\\3}}&&]]>\text{<acroref type="theorem" acro="BRS" />}
+<![CDATA[&=\rsp{L}&&]]><acroref type="theorem" acro="FS" />\\
+<![CDATA[&=\spn{\colvector{1\\0\\-1\\-1},\,\colvector{0\\1\\1\\3}}&&]]><acroref type="theorem" acro="BRS" />
 </alignmath>
 </solution>
 </exercise>

File src/section-ILT.xml

 <![CDATA[0&0\\0&0]]>
 \end{bmatrix}\\
 <![CDATA[&=\lt{T}{a_1+b_1x+c_1x^2+d_1x^3}-\lt{T}{a_2+b_2x+c_2x^2+d_2x^3}&&\text{Hypothesis}\\]]>
-<![CDATA[&=\lt{T}{(a_1+b_1x+c_1x^2+d_1x^3)-(a_2+b_2x+c_2x^2+d_2x^3)}&&]]>\text{<acroref type="definition" acro="LT" />}\\
+<![CDATA[&=\lt{T}{(a_1+b_1x+c_1x^2+d_1x^3)-(a_2+b_2x+c_2x^2+d_2x^3)}&&]]><acroref type="definition" acro="LT" />\\
 <![CDATA[&=\lt{T}{(a_1-a_2)+(b_1-b_2)x+(c_1-c_2)x^2+(d_1-d_2)x^3}&&\text{Operations in $P_3$}\\]]>
 <![CDATA[&=]]>
 \begin{bmatrix}
 
 <p>Suppose we assume that $\vect{x},\,\vect{y}\in\krn{T}$.  Is $\vect{x}+\vect{y}\in\krn{T}$?
 <alignmath>
-<![CDATA[\lt{T}{\vect{x}+\vect{y}}&=\lt{T}{\vect{x}}+\lt{T}{\vect{y}}&&]]>\text{<acroref type="definition" acro="LT" />}\\
+<![CDATA[\lt{T}{\vect{x}+\vect{y}}&=\lt{T}{\vect{x}}+\lt{T}{\vect{y}}&&]]><acroref type="definition" acro="LT" />\\
 <![CDATA[&=\zerovector+\zerovector&&\vect{x},\,\vect{y}\in\krn{T}\\]]>
-<![CDATA[&=\zerovector&&]]>\text{<acroref type="property" acro="Z" />}
+<![CDATA[&=\zerovector&&]]><acroref type="property" acro="Z" />
 </alignmath></p>
 
 <p>This qualifies $\vect{x}+\vect{y}$ for membership in $\krn{T}$.  So we have additive closure.</p>
 
 <p>Suppose we assume that $\alpha\in\complex{\null}$ and $\vect{x}\in\krn{T}$.  Is $\alpha\vect{x}\in\krn{T}$?
 <alignmath>
-<![CDATA[\lt{T}{\alpha\vect{x}}&=\alpha\lt{T}{\vect{x}}&&]]>\text{<acroref type="definition" acro="LT" />}\\
+<![CDATA[\lt{T}{\alpha\vect{x}}&=\alpha\lt{T}{\vect{x}}&&]]><acroref type="definition" acro="LT" />\\
 <![CDATA[&=\alpha\zerovector&&\vect{x}\in\krn{T}\\]]>
-<![CDATA[&=\zerovector&&]]>\text{<acroref type="theorem" acro="ZVSM" />}
+<![CDATA[&=\zerovector&&]]><acroref type="theorem" acro="ZVSM" />
 </alignmath>
 </p>
 
 <p>Let $M=\setparts{\vect{u}+\vect{z}}{\vect{z}\in\krn{T}}$.  First, we show that $M\subseteq\preimage{T}{\vect{v}}$.  Suppose that $\vect{w}\in M$, so $\vect{w}$ has the form $\vect{w}=\vect{u}+\vect{z}$, where $\vect{z}\in\krn{T}$.  Then
 <alignmath>
 <![CDATA[\lt{T}{\vect{w}}&=\lt{T}{\vect{u}+\vect{z}}\\]]>
-<![CDATA[&=\lt{T}{\vect{u}}+\lt{T}{\vect{z}}&&]]>\text{<acroref type="definition" acro="LT" />}\\
+<![CDATA[&=\lt{T}{\vect{u}}+\lt{T}{\vect{z}}&&]]><acroref type="definition" acro="LT" />\\
 <![CDATA[&=\vect{v}+\zerovector&&\vect{u}\in\preimage{T}{\vect{v}},\ \vect{z}\in\krn{T}\\]]>
-<![CDATA[&=\vect{v}&&]]>\text{<acroref type="property" acro="Z" />}
+<![CDATA[&=\vect{v}&&]]><acroref type="property" acro="Z" />
 </alignmath>
 which qualifies $\vect{w}$ for membership in the preimage of $\vect{v}$, $\vect{w}\in\preimage{T}{\vect{v}}$.</p>
 
 <p>For the opposite inclusion, suppose $\vect{x}\in\preimage{T}{\vect{v}}$.  Then,
 <alignmath>
-<![CDATA[\lt{T}{\vect{x}-\vect{u}}&=\lt{T}{\vect{x}}-\lt{T}{\vect{u}}&&]]>\text{<acroref type="definition" acro="LT" />}\\
+<![CDATA[\lt{T}{\vect{x}-\vect{u}}&=\lt{T}{\vect{x}}-\lt{T}{\vect{u}}&&]]><acroref type="definition" acro="LT" />\\
 <![CDATA[&=\vect{v}-\vect{v}&&\vect{x},\,\vect{u}\in\preimage{T}{\vect{v}}\\]]>
 <![CDATA[&=\zerovector]]>
 </alignmath>
 <p><implyforward /> We assume $T$ is injective and we need to establish that two sets are equal (<acroref type="definition" acro="SE" />).  Since the kernel is a subspace (<acroref type="theorem" acro="KLTS" />), $\set{\zerovector}\subseteq\krn{T}$.  To establish the opposite inclusion, suppose $\vect{x}\in\krn{T}$.
 <alignmath>
 \lt{T}{\vect{x}}
-<![CDATA[&=\zerovector&&]]>\text{<acroref type="definition" acro="KLT" />}\\
+<![CDATA[&=\zerovector&&]]><acroref type="definition" acro="KLT" />\\
 <![CDATA[&=\lt{T}{\zerovector}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="LTTZZ" />}
+<![CDATA[&&]]><acroref type="theorem" acro="LTTZZ" />
 </alignmath>
 </p>
 
 <alignmath>
 \lt{T}{\vect{x}-\vect{y}}
 <![CDATA[&=\lt{T}{\vect{x}}-\lt{T}{\vect{y}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="LT" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="LT" />\\
 <![CDATA[&=\zerovector]]>
 <![CDATA[&&\text{Hypothesis}]]>
 </alignmath>
 which you can check is an element of $\krn{T}$ for <acroref type="archetype" acro="Q" />.  Choose a vector $\vect{x}$ at random, and then compute $\vect{y}=\vect{x}+\vect{z}$ (verify this computation back in <acroref type="example" acro="NIAQ" />).  Then
 <alignmath>
 <![CDATA[\lt{T}{\vect{y}}&=\lt{T}{\vect{x}+\vect{z}}\\]]>
-<![CDATA[&=\lt{T}{\vect{x}}+\lt{T}{\vect{z}}&&]]>\text{<acroref type="definition" acro="LT" />}\\
+<![CDATA[&=\lt{T}{\vect{x}}+\lt{T}{\vect{z}}&&]]><acroref type="definition" acro="LT" />\\
 <![CDATA[&=\lt{T}{\vect{x}}+\zerovector&&\vect{z}\in\krn{T}\\]]>
-<![CDATA[&=\lt{T}{\vect{x}}&&]]>\text{<acroref type="property" acro="Z" />}
+<![CDATA[&=\lt{T}{\vect{x}}&&]]><acroref type="property" acro="Z" />
 </alignmath>
 </p>
 
 <p>Begin with a relation of linear dependence on $R$ (<acroref type="definition" acro="RLD" />, <acroref type="definition" acro="LI" />),
 <alignmath>
 <![CDATA[a_1\lt{T}{\vect{u}_1}+a_2\lt{T}{\vect{u}_2}+a_3\lt{T}{\vect{u}_3}+\ldots+a_t\lt{T}{\vect{u}_t}&=\zerovector\\]]>
-<![CDATA[\lt{T}{\lincombo{a}{u}{t}}&=\zerovector&&]]>\text{<acroref type="theorem" acro="LTLC" />}\\
-<![CDATA[\lincombo{a}{u}{t}&\in\krn{T}&&]]>\text{<acroref type="definition" acro="KLT" />}\\
-<![CDATA[\lincombo{a}{u}{t}&\in\set{\zerovector}&&]]>\text{<acroref type="theorem" acro="KILT" />}\\
-<![CDATA[\lincombo{a}{u}{t}&=\zerovector&&]]>\text{<acroref type="definition" acro="SET" />}\\
+<![CDATA[\lt{T}{\lincombo{a}{u}{t}}&=\zerovector&&]]><acroref type="theorem" acro="LTLC" />\\
+<![CDATA[\lincombo{a}{u}{t}&\in\krn{T}&&]]><acroref type="definition" acro="KLT" />\\
+<![CDATA[\lincombo{a}{u}{t}&\in\set{\zerovector}&&]]><acroref type="theorem" acro="KILT" />\\
+<![CDATA[\lincombo{a}{u}{t}&=\zerovector&&]]><acroref type="definition" acro="SET" />\\
 </alignmath>
 </p>
 
 <alignmath>
 \zerovector
 <![CDATA[&=\lt{T}{\vect{u}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="KLT" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="KLT" />\\
 <![CDATA[&=\lt{T}{\lincombo{a}{u}{m}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="SSVS" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="SSVS" />\\
 <![CDATA[&=a_1\lt{T}{\vect{u}_1}+a_2\lt{T}{\vect{u}_2}+a_3\lt{T}{\vect{u}_3}+\cdots+a_m\lt{T}{\vect{u}_m}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="LTLC" />}
+<![CDATA[&&]]><acroref type="theorem" acro="LTLC" />
 </alignmath>
 </p>
 
 <p>This is a relation of linear dependence (<acroref type="definition" acro="RLD" />) on the linearly independent set $C$, so the scalars are all zero:  $a_1=a_2=a_3=\cdots=a_m=0$.  Then
 <alignmath>
 <![CDATA[\vect{u}&=\lincombo{a}{u}{m}\\]]>
-<![CDATA[&=0\vect{u}_1+0\vect{u}_2+0\vect{u}_3+\cdots+0\vect{u}_m&&]]>\text{<acroref type="theorem" acro="ZSSM" />}\\
-<![CDATA[&=\zerovector+\zerovector+\zerovector+\cdots+\zerovector&&]]>\text{<acroref type="theorem" acro="ZSSM" />}\\
-<![CDATA[&=\zerovector&&]]>\text{<acroref type="property" acro="Z" />}
+<![CDATA[&=0\vect{u}_1+0\vect{u}_2+0\vect{u}_3+\cdots+0\vect{u}_m&&]]><acroref type="theorem" acro="ZSSM" />\\
+<![CDATA[&=\zerovector+\zerovector+\zerovector+\cdots+\zerovector&&]]><acroref type="theorem" acro="ZSSM" />\\
+<![CDATA[&=\zerovector&&]]><acroref type="property" acro="Z" />
 </alignmath>
 </p>
 
 <p>That the composition is a linear transformation was established in <acroref type="theorem" acro="CLTLT" />, so we need only establish that the composition is injective.  Applying <acroref type="definition" acro="ILT" />, choose $\vect{x}$, $\vect{y}$ from $U$.  Then if $\lt{\left(\compose{S}{T}\right)}{\vect{x}}=\lt{\left(\compose{S}{T}\right)}{\vect{y}}$,
 <alignmath>
 <![CDATA[&\Rightarrow&\lt{S}{\lt{T}{\vect{x}}}&=\lt{S}{\lt{T}{\vect{y}}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="LTC" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="LTC" />\\
 <![CDATA[&\Rightarrow&\lt{T}{\vect{x}}&=\lt{T}{\vect{y}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ILT" /> for $S$}\\
+<![CDATA[&&]]><acroref type="definition" acro="ILT" /> for $S$\\
 <![CDATA[&\Rightarrow&\vect{x}&=\vect{y}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="ILT" /> for $T$}
+<![CDATA[&&]]><acroref type="definition" acro="ILT" /> for $T$
 </alignmath>
 </p>
 
 </problem>
 <solution contributor="robertbeezer">We are asked to prove that $\krn{T}$ is a subset of $\krn{\compose{S}{T}}$.  Employing <acroref type="definition" acro="SSET" />, choose $\vect{x}\in\krn{T}$.  Then we know that $\lt{T}{\vect{x}}=\zerovector$.  So
 <alignmath>
-<![CDATA[\lt{\left(\compose{S}{T}\right)}{\vect{x}}&=\lt{S}{\lt{T}{\vect{x}}}&&]]>\text{<acroref type="definition" acro="LTC" />}\\
+<![CDATA[\lt{\left(\compose{S}{T}\right)}{\vect{x}}&=\lt{S}{\lt{T}{\vect{x}}}&&]]><acroref type="definition" acro="LTC" />\\
 <![CDATA[&=\lt{S}{\zerovector}&&\vect{x}\in\krn{T}\\]]>
-<![CDATA[&=\zerovector&&]]>\text{<acroref type="theorem" acro="LTTZZ" />}
+<![CDATA[&=\zerovector&&]]><acroref type="theorem" acro="LTTZZ" />
 </alignmath>
 This qualifies $\vect{x}$ for membership in $\krn{\compose{S}{T}}$.
 </solution>

File src/section-IVLT.xml

 <alignmath>
 <![CDATA[\lt{\ltinverse{T}}{\vect{x}+\vect{y}}&=]]>
 \lt{\ltinverse{T}}{\lt{T}{\lt{\ltinverse{T}}{\vect{x}}}+\lt{T}{\lt{\ltinverse{T}}{\vect{y}}}}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
-<![CDATA[&=\lt{\ltinverse{T}}{\lt{T}{\lt{\ltinverse{T}}{\vect{x}}+\lt{\ltinverse{T}}{\vect{y}}}}&&]]>\text{<acroref type="definition" acro="LT" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="IVLT" />\\
+<![CDATA[&=\lt{\ltinverse{T}}{\lt{T}{\lt{\ltinverse{T}}{\vect{x}}+\lt{\ltinverse{T}}{\vect{y}}}}&&]]><acroref type="definition" acro="LT" />\\
 <![CDATA[&=\lt{\ltinverse{T}}{\vect{x}}+\lt{\ltinverse{T}}{\vect{y}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="IVLT" />}
+<![CDATA[&&]]><acroref type="definition" acro="IVLT" />
 </alignmath>
 </p>
 
 <alignmath>
 <![CDATA[\lt{\ltinverse{T}}{\alpha\vect{x}}&=]]>
 \lt{\ltinverse{T}}{\alpha\lt{T}{\lt{\ltinverse{T}}{\vect{x}}}}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="IVLT" />\\
 <![CDATA[&=\lt{\ltinverse{T}}{\lt{T}{\alpha\lt{\ltinverse{T}}{\vect{x}}}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="LT" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="LT" />\\
 <![CDATA[&=\alpha\lt{\ltinverse{T}}{\vect{x}}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="IVLT" />}
+<![CDATA[&&]]><acroref type="definition" acro="IVLT" />
 </alignmath>
 </p>
 
 <p><implyforward />  Since $T$ is presumed invertible, we can employ its inverse, $\ltinverse{T}$ (<acroref type="definition" acro="IVLT" />).  To see that $T$ is injective, suppose $\vect{x},\,\vect{y}\in U$ and assume that $\lt{T}{\vect{x}}=\lt{T}{\vect{y}}$,
 <alignmath>
 \vect{x}
-<![CDATA[&=\lt{I_U}{\vect{x}}&&]]>\text{<acroref type="definition" acro="IDLT" />}\\
-<![CDATA[&=\lt{\left(\compose{\ltinverse{T}}{T}\right)}{\vect{x}}&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
-<![CDATA[&=\lt{\ltinverse{T}}{\lt{T}{\vect{x}}}&&]]>\text{<acroref type="definition" acro="LTC" />}\\
-<![CDATA[&=\lt{\ltinverse{T}}{\lt{T}{\vect{y}}}&&]]>\text{<acroref type="definition" acro="ILT" />}\\
-<![CDATA[&=\lt{\left(\compose{\ltinverse{T}}{T}\right)}{\vect{y}}&&]]>\text{<acroref type="definition" acro="LTC" />}\\
-<![CDATA[&=\lt{I_U}{\vect{y}}&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
-<![CDATA[&=\vect{y}&&]]>\text{<acroref type="definition" acro="IDLT" />}
+<![CDATA[&=\lt{I_U}{\vect{x}}&&]]><acroref type="definition" acro="IDLT" />\\
+<![CDATA[&=\lt{\left(\compose{\ltinverse{T}}{T}\right)}{\vect{x}}&&]]><acroref type="definition" acro="IVLT" />\\
+<![CDATA[&=\lt{\ltinverse{T}}{\lt{T}{\vect{x}}}&&]]><acroref type="definition" acro="LTC" />\\
+<![CDATA[&=\lt{\ltinverse{T}}{\lt{T}{\vect{y}}}&&]]><acroref type="definition" acro="ILT" />\\
+<![CDATA[&=\lt{\left(\compose{\ltinverse{T}}{T}\right)}{\vect{y}}&&]]><acroref type="definition" acro="LTC" />\\
+<![CDATA[&=\lt{I_U}{\vect{y}}&&]]><acroref type="definition" acro="IVLT" />\\
+<![CDATA[&=\vect{y}&&]]><acroref type="definition" acro="IDLT" />
 </alignmath>
 So by <acroref type="definition" acro="ILT" /> $T$ is injective.</p>
 
 <p>To check that $T$ is surjective, suppose $\vect{v}\in V$.  Then $\lt{\ltinverse{T}}{\vect{v}}$ is a vector in $U$.  Compute
 <alignmath>
 \lt{T}{\lt{\ltinverse{T}}{\vect{v}}}
-<![CDATA[&=\lt{\left(\compose{T}{\ltinverse{T}}\right)}{\vect{v}}&&]]>\text{<acroref type="definition" acro="LTC" />}\\
-<![CDATA[&=\lt{I_V}{\vect{v}}&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
-<![CDATA[&=\vect{v}&&]]>\text{<acroref type="definition" acro="IDLT" />}
+<![CDATA[&=\lt{\left(\compose{T}{\ltinverse{T}}\right)}{\vect{v}}&&]]><acroref type="definition" acro="LTC" />\\
+<![CDATA[&=\lt{I_V}{\vect{v}}&&]]><acroref type="definition" acro="IVLT" />\\
+<![CDATA[&=\vect{v}&&]]><acroref type="definition" acro="IDLT" />
 </alignmath>
 So there is an element from $U$, when used as an input to $T$ (namely $\lt{\ltinverse{T}}{\vect{v}}$) that produces the desired output, $\vect{v}$, and hence $T$ is surjective by <acroref type="definition" acro="SLT" />.</p>
 
 \lt{\ltinverse{T}}{p+qx+rx^2}
 <![CDATA[&=]]>
 \lt{\ltinverse{T}}{p(1)+q(x)+r(x^2)}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="VRRB" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="VRRB" />\\
 <![CDATA[&=]]>
 p\lt{\ltinverse{T}}{1}+
 q\lt{\ltinverse{T}}{x}+
 r\lt{\ltinverse{T}}{x^2}
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="LTLC" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="LTLC" />\\
 <![CDATA[&=]]>
 <![CDATA[p\begin{bmatrix}-6&10\\10&-3\end{bmatrix}+]]>
 <![CDATA[q\begin{bmatrix}-3&4\\4&-1\end{bmatrix}+]]>
 <alignmath>
 \lt{\left(\compose{\left(\compose{S}{T}\right)}{\left(\compose{\ltinverse{T}}{\ltinverse{S}}\right)}\right)}{\vect{w}}
 <![CDATA[&=\lt{S}{\lt{T}{\lt{\ltinverse{T}}{\lt{\ltinverse{S}}{\vect{w}}}}}\\]]>
-<![CDATA[&=\lt{S}{\lt{I_V}{\lt{\ltinverse{S}}{\vect{w}}}}&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
-<![CDATA[&=\lt{S}{\lt{\ltinverse{S}}{\vect{w}}}&&]]>\text{<acroref type="definition" acro="IDLT" />}\\
-<![CDATA[&=\vect{w}&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
-<![CDATA[&=\lt{I_W}{\vect{w}}&&]]>\text{<acroref type="definition" acro="IDLT" />}
+<![CDATA[&=\lt{S}{\lt{I_V}{\lt{\ltinverse{S}}{\vect{w}}}}&&]]><acroref type="definition" acro="IVLT" />\\
+<![CDATA[&=\lt{S}{\lt{\ltinverse{S}}{\vect{w}}}&&]]><acroref type="definition" acro="IDLT" />\\
+<![CDATA[&=\vect{w}&&]]><acroref type="definition" acro="IVLT" />\\
+<![CDATA[&=\lt{I_W}{\vect{w}}&&]]><acroref type="definition" acro="IDLT" />
 </alignmath>
 </p>
 
 <alignmath>
 \lt{\left(\compose{\left(\compose{\ltinverse{T}}{\ltinverse{S}}\right)}{\left(\compose{S}{T}\right)}\right)}{\vect{u}}
 <![CDATA[&=\lt{\ltinverse{T}}{\lt{\ltinverse{S}}{\lt{S}{\lt{T}{\vect{u}}}}}\\]]>
-<![CDATA[&=\lt{\ltinverse{T}}{\lt{I_V}{\lt{T}{\vect{u}}}}&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
-<![CDATA[&=\lt{\ltinverse{T}}{\lt{T}{\vect{u}}}&&]]>\text{<acroref type="definition" acro="IDLT" />}\\
-<![CDATA[&=\vect{u}&&]]>\text{<acroref type="definition" acro="IVLT" />}\\
-<![CDATA[&=\lt{I_U}{\vect{u}}&&]]>\text{<acroref type="definition" acro="IDLT" />}
+<![CDATA[&=\lt{\ltinverse{T}}{\lt{I_V}{\lt{T}{\vect{u}}}}&&]]><acroref type="definition" acro="IVLT" />\\
+<![CDATA[&=\lt{\ltinverse{T}}{\lt{T}{\vect{u}}}&&]]><acroref type="definition" acro="IDLT" />\\
+<![CDATA[&=\vect{u}&&]]><acroref type="definition" acro="IVLT" />\\
+<![CDATA[&=\lt{I_U}{\vect{u}}&&]]><acroref type="definition" acro="IDLT" />
 </alignmath>
 so $\compose{\left(\compose{\ltinverse{T}}{\ltinverse{S}}\right)}{\left(\compose{S}{T}\right)}=I_U$.</p>
 
 <p>Rather than doing it straight-away (which is very easy), we will apply the transformation $T$ to convert into a linear combination of matrices, and then compute in $M_{22}$ according to the definitions of the vector space operations there (<acroref type="example" acro="VSM" />),
 <alignmath>
 <![CDATA[&\lt{T}{5(2+3x-4x^2+5x^3)+(-3)(3-5x+3x^2+x^3)}\\]]>
-<![CDATA[&=5\lt{T}{2+3x-4x^2+5x^3}+(-3)\lt{T}{3-5x+3x^2+x^3}&&]]>\text{<acroref type="theorem" acro="LTLC" />}\\
+<![CDATA[&=5\lt{T}{2+3x-4x^2+5x^3}+(-3)\lt{T}{3-5x+3x^2+x^3}&&]]><acroref type="theorem" acro="LTLC" />\\
 <![CDATA[&=5]]>
 \begin{bmatrix}
 <![CDATA[5 & 10\\5 & -2]]>
 
 <p>Then
 <alignmath>
-<![CDATA[\zerovector&=\lt{T}{\zerovector}&&]]>\text{<acroref type="theorem" acro="LTTZZ" />}\\
+<![CDATA[\zerovector&=\lt{T}{\zerovector}&&]]><acroref type="theorem" acro="LTTZZ" />\\
 <![CDATA[&=T\left(\lincombo{a}{u}{s}+\right.\\]]>
-<![CDATA[&\quad\quad\quad\quad\left.\lincombo{b}{w}{r}\right)&&]]>\text{<acroref type="definition" acro="LI" />}\\
+<![CDATA[&\quad\quad\quad\quad\left.\lincombo{b}{w}{r}\right)&&]]><acroref type="definition" acro="LI" />\\
 <![CDATA[&=a_1\lt{T}{\vect{u}_1}+a_2\lt{T}{\vect{u}_2}+a_3\lt{T}{\vect{u}_3}+\cdots+a_s\lt{T}{\vect{u}_s}+\\]]>
 <![CDATA[&\quad\quad b_1\lt{T}{\vect{w}_1}+b_2\lt{T}{\vect{w}_2}+b_3\lt{T}{\vect{w}_3}+\cdots+b_r\lt{T}{\vect{w}_r}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="LTLC" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="LTLC" />\\
 <![CDATA[&=a_1\zerovector+a_2\zerovector+a_3\zerovector+\cdots+a_s\zerovector+\\]]>
 <![CDATA[&\quad\quad b_1\lt{T}{\vect{w}_1}+b_2\lt{T}{\vect{w}_2}+b_3\lt{T}{\vect{w}_3}+\cdots+b_r\lt{T}{\vect{w}_r}]]>
-<![CDATA[&&]]>\text{<acroref type="definition" acro="KLT" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="KLT" />\\
 <![CDATA[&=\zerovector+\zerovector+\zerovector+\cdots+\zerovector+\\]]>
 <![CDATA[&\quad\quad b_1\lt{T}{\vect{w}_1}+b_2\lt{T}{\vect{w}_2}+b_3\lt{T}{\vect{w}_3}+\cdots+b_r\lt{T}{\vect{w}_r}]]>
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="ZVSM" />}\\
-<![CDATA[&=b_1\lt{T}{\vect{w}_1}+b_2\lt{T}{\vect{w}_2}+b_3\lt{T}{\vect{w}_3}+\cdots+b_r\lt{T}{\vect{w}_r}&&]]>\text{<acroref type="property" acro="Z" />}\\
-<![CDATA[&=b_1\vect{v}_1+b_2\vect{v}_2+b_3\vect{v}_3+\cdots+b_r\vect{v}_r&&]]>\text{<acroref type="definition" acro="PI" />}
+<![CDATA[&&]]><acroref type="theorem" acro="ZVSM" />\\
+<![CDATA[&=b_1\lt{T}{\vect{w}_1}+b_2\lt{T}{\vect{w}_2}+b_3\lt{T}{\vect{w}_3}+\cdots+b_r\lt{T}{\vect{w}_r}&&]]><acroref type="property" acro="Z" />\\
+<![CDATA[&=b_1\vect{v}_1+b_2\vect{v}_2+b_3\vect{v}_3+\cdots+b_r\vect{v}_r&&]]><acroref type="definition" acro="PI" />
 </alignmath>
 </p>
 
 <p>This is a relation of linear dependence on $R$ (<acroref type="definition" acro="RLD" />), and since $R$ is a linearly independent set (<acroref type="definition" acro="LI" />), we see that $b_1=b_2=b_3=\ldots=b_r=0$.  Then the original relation of linear dependence on $B$ becomes
 <alignmath>
 <![CDATA[\zerovector&=\lincombo{a}{u}{s}+0\vect{w}_1+0\vect{w}_2+\ldots+0\vect{w}_r\\]]>
-<![CDATA[&=\lincombo{a}{u}{s}+\zerovector+\zerovector+\ldots+\zerovector&&]]>\text{<acroref type="theorem" acro="ZSSM" />}\\
-<![CDATA[&=\lincombo{a}{u}{s}&&]]>\text{<acroref type="property" acro="Z" />}
+<![CDATA[&=\lincombo{a}{u}{s}+\zerovector+\zerovector+\ldots+\zerovector&&]]><acroref type="theorem" acro="ZSSM" />\\
+<![CDATA[&=\lincombo{a}{u}{s}&&]]><acroref type="property" acro="Z" />
 </alignmath>
 </p>
 
 
 <p>Then
 <alignmath>
-<![CDATA[\lt{T}{\vect{u}-\vect{y}}&=\lt{T}{\vect{u}}-\lt{T}{\vect{y}}&&]]>\text{<acroref type="theorem" acro="LTLC" />}\\
+<![CDATA[\lt{T}{\vect{u}-\vect{y}}&=\lt{T}{\vect{u}}-\lt{T}{\vect{y}}&&]]><acroref type="theorem" acro="LTLC" />\\
 <![CDATA[&=\lt{T}{\vect{u}}-\lt{T}{\lincombo{c}{w}{r}}&&\text{Substitution}\\]]>
-<![CDATA[&=\lt{T}{\vect{u}}-\left(c_1\lt{T}{\vect{w}_1}+c_2\lt{T}{\vect{w}_2}+\cdots+c_r\lt{T}{\vect{w}_r}\right)&&]]>\text{<acroref type="theorem" acro="LTLC" />}\\
+<![CDATA[&=\lt{T}{\vect{u}}-\left(c_1\lt{T}{\vect{w}_1}+c_2\lt{T}{\vect{w}_2}+\cdots+c_r\lt{T}{\vect{w}_r}\right)&&]]><acroref type="theorem" acro="LTLC" />\\
 <![CDATA[&=\lt{T}{\vect{u}}-\left(\lincombo{c}{v}{r}\right)&&\vect{w}_i\in\preimage{T}{\vect{v}_i}\\]]>
 <![CDATA[&=\lt{T}{\vect{u}}-\lt{T}{\vect{u}}&&\text{Substitution}\\]]>
-<![CDATA[&=\zerovector&&]]>\text{<acroref type="property" acro="AI" />}
+<![CDATA[&=\zerovector&&]]><acroref type="property" acro="AI" />
 </alignmath>
 </p>
 
 
 <p>These two archetypes each have three equations in four variables, so either the resulting linear systems are inconsistent, or they are consistent and application of <acroref type="theorem" acro="CMVEI" /> tells us that the system has infinitely many solutions.  Considering these same parameters for the linear transformation, the dimension of the domain, $\complex{4}$, is four, while the codomain, $\complex{3}$, has dimension three.  Then
 <alignmath>
-<![CDATA[\nullity{T}&=\dimension{\complex{4}}-\rank{T}&&]]>\text{<acroref type="theorem" acro="RPNDD" />}\\
-<![CDATA[&=4-\dimension{\rng{T}}&&]]>\text{<acroref type="definition" acro="ROLT" />}\\
+<![CDATA[\nullity{T}&=\dimension{\complex{4}}-\rank{T}&&]]><acroref type="theorem" acro="RPNDD" />\\
+<![CDATA[&=4-\dimension{\rng{T}}&&]]><acroref type="definition" acro="ROLT" />\\
 <![CDATA[&\geq 4-3&&\text{$\rng{T}$ subspace of $\complex{3}$}\\]]>
 <![CDATA[&=1]]>
 </alignmath>
 (b)  We can establish that $S$ is surjective by considering the rank and nullity of $S$.
 <alignmath>
 \rank{S}
-<![CDATA[&=\dimension{P_3}-\nullity{S}&&]]>\text{<acroref type="theorem" acro="RPNDD" />}\\
+<![CDATA[&=\dimension{P_3}-\nullity{S}&&]]><acroref type="theorem" acro="RPNDD" />\\
 <![CDATA[&=4-0\\]]>
 <![CDATA[&=\dimension{M_{22}}]]>
 </alignmath>
 <solution contributor="robertbeezer">Since $\compose{S}{T}$ is invertible, by <acroref type="theorem" acro="ILTIS" /> $\compose{S}{T}$ is injective and therefore has a trivial kernel by <acroref type="theorem" acro="KILT" />.  Then
 <alignmath>
 \krn{T}
-<![CDATA[&\subseteq\krn{\compose{S}{T}}&&]]>\text{<acroref type="exercise" acro="ILT.T15" />}\\
-<![CDATA[&=\set{\zerovector}&&]]>\text{<acroref type="theorem" acro="KILT" />}
+<![CDATA[&\subseteq\krn{\compose{S}{T}}&&]]><acroref type="exercise" acro="ILT.T15" />\\
+<![CDATA[&=\set{\zerovector}&&]]><acroref type="theorem" acro="KILT" />
 </alignmath>
 Since $T$ has a trivial kernel, by <acroref type="theorem" acro="KILT" />, $T$ is injective.  Also,
 <alignmath>
 \rank{T}
-<![CDATA[&=\dimension{U}-\nullity{T}&&]]>\text{<acroref type="theorem" acro="RPNDD" />}\\
-<![CDATA[&=\dimension{U}-0&&]]>\text{<acroref type="theorem" acro="NOILT" />}\\
+<![CDATA[&=\dimension{U}-\nullity{T}&&]]><acroref type="theorem" acro="RPNDD" />\\
+<![CDATA[&=\dimension{U}-0&&]]><acroref type="theorem" acro="NOILT" />\\
 <![CDATA[&=\dimension{V}&&\text{Hypothesis}]]>
 </alignmath>
 Since $\rng{T}\subseteq V$, <acroref type="theorem" acro="EDYES" /> gives $\rng{T}=V$, so by <acroref type="theorem" acro="RSLT" />, $T$ is surjective.  Finally, by <acroref type="theorem" acro="ILTIS" />, $T$ is invertible.<br /><br />
 Since $\compose{S}{T}$ is invertible, by <acroref type="theorem" acro="ILTIS" /> $\compose{S}{T}$ is surjective and therefore has a full range by <acroref type="theorem" acro="RSLT" />.  Then
 <alignmath>
 W
-<![CDATA[&=\rng{\compose{S}{T}}&&]]>\text{<acroref type="theorem" acro="RSLT" />}\\
-<![CDATA[&\subseteq\rng{S}&&]]>\text{<acroref type="exercise" acro="SLT.T15" />}
+<![CDATA[&=\rng{\compose{S}{T}}&&]]><acroref type="theorem" acro="RSLT" />\\
+<![CDATA[&\subseteq\rng{S}&&]]><acroref type="exercise" acro="SLT.T15" />
 </alignmath>
 Since $\rng{S}\subseteq W$ we have $\rng{S}=W$ and by <acroref type="theorem" acro="RSLT" />, $S$ is surjective.  By an application of <acroref type="theorem" acro="RPNDD" /> similar to the first part of this solution, we see that $S$ has a trivial kernel, is therefore injective (<acroref type="theorem" acro="KILT" />), and thus invertible (<acroref type="theorem" acro="ILTIS" />).
 </solution>

File src/section-LC.xml

 <p><implyreverse />   Suppose we have the vector equality between $\vect{b}$ and the linear combination of the columns of $A$.  Then for $1\leq i\leq m$,
 <alignmath>
 b_i
-<![CDATA[&=\vectorentry{\vect{b}}{i}&&]]>\text{<acroref type="definition" acro="CV" />}\\
+<![CDATA[&=\vectorentry{\vect{b}}{i}&&]]><acroref type="definition" acro="CV" />\\
 <![CDATA[&=\vectorentry{]]>
 \vectorentry{\vect{x}}{1}\vect{A}_1+
 \vectorentry{\vect{x}}{2}\vect{A}_2+
 \vectorentry{\vectorentry{\vect{x}}{3}\vect{A}_3}{i}+
 \cdots+
 \vectorentry{\vectorentry{\vect{x}}{n}\vect{A}_n}{i}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="CVA" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="CVA" />\\
 <![CDATA[&=]]>
 \vectorentry{\vect{x}}{1}\vectorentry{\vect{A}_1}{i}+
 \vectorentry{\vect{x}}{2}\vectorentry{\vect{A}_2}{i}+
 \vectorentry{\vect{x}}{3}\vectorentry{\vect{A}_3}{i}+
 \cdots+
 \vectorentry{\vect{x}}{n}\vectorentry{\vect{A}_n}{i}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="CVSM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="CVSM" />\\
 <![CDATA[&=]]>
 \vectorentry{\vect{x}}{1}a_{i1}+
 \vectorentry{\vect{x}}{2}a_{i2}+
 \vectorentry{\vect{x}}{3}a_{i3}+
 \cdots+
 \vectorentry{\vect{x}}{n}a_{in}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="CV" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="CV" />\\
 <![CDATA[&=]]>
 a_{i1}\vectorentry{\vect{x}}{1}+
 a_{i2}\vectorentry{\vect{x}}{2}+
 a_{i3}\vectorentry{\vect{x}}{3}+
 \cdots+
 a_{in}\vectorentry{\vect{x}}{n}
-<![CDATA[&&]]>\text{<acroref type="property" acro="CMCN" />}
+<![CDATA[&&]]><acroref type="property" acro="CMCN" />
 </alignmath></p>
 
 <p>This says that the entries of $\vect{x}$ form a solution to equation $i$ of $\linearsystem{A}{\vect{b}}$ for all $1\leq i\leq m$, in other words, $\vect{x}$ is a solution to $\linearsystem{A}{\vect{b}}$.</p>
 <p><implyforward />  Suppose now that $\vect{x}$ is a solution to the linear system $\linearsystem{A}{\vect{b}}$.  Then for all $1\leq i\leq m$,
 <alignmath>
 \vectorentry{\vect{b}}{i}
-<![CDATA[&=b_i&&]]>\text{<acroref type="definition" acro="CV" />}\\
+<![CDATA[&=b_i&&]]><acroref type="definition" acro="CV" />\\
 <![CDATA[&=]]>
 a_{i1}\vectorentry{\vect{x}}{1}+
 a_{i2}\vectorentry{\vect{x}}{2}+
 \vectorentry{\vect{x}}{3}a_{i3}+
 \cdots+
 \vectorentry{\vect{x}}{n}a_{in}
-<![CDATA[&&]]>\text{<acroref type="property" acro="CMCN" />}\\
+<![CDATA[&&]]><acroref type="property" acro="CMCN" />\\
 <![CDATA[&=]]>
 \vectorentry{\vect{x}}{1}\vectorentry{\vect{A}_1}{i}+
 \vectorentry{\vect{x}}{2}\vectorentry{\vect{A}_2}{i}+
 \vectorentry{\vect{x}}{3}\vectorentry{\vect{A}_3}{i}+
 \cdots+
 \vectorentry{\vect{x}}{n}\vectorentry{\vect{A}_n}{i}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="CV" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="CV" />\\
 <![CDATA[&=]]>
 \vectorentry{\vectorentry{\vect{x}}{1}\vect{A}_1}{i}+
 \vectorentry{\vectorentry{\vect{x}}{2}\vect{A}_2}{i}+
 \vectorentry{\vectorentry{\vect{x}}{3}\vect{A}_3}{i}+
 \cdots+
 \vectorentry{\vectorentry{\vect{x}}{n}\vect{A}_n}{i}
-<![CDATA[&&]]>\text{<acroref type="definition" acro="CVSM" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="CVSM" />\\
 <![CDATA[&=]]>\vectorentry{
 \vectorentry{\vect{x}}{1}\vect{A}_1+
 \vectorentry{\vect{x}}{2}\vect{A}_2+
 \vectorentry{\vect{x}}{3}\vect{A}_3+
 \cdots+
 \vectorentry{\vect{x}}{n}\vect{A}_n
-}{i}<![CDATA[&&]]>\text{<acroref type="definition" acro="CVA" />}
+}{i}<![CDATA[&&]]><acroref type="definition" acro="CVA" />
 </alignmath>
 Since the components of $\vect{b}$ and the linear combination of the columns of $A$ agree for all $1\leq i\leq m$, <acroref type="definition" acro="CVE" /> tells us that the vectors are equal.</p>
 
 <intertext>Now we will use the definitions of column vector addition and scalar multiplication to express this vector as a linear combination,</intertext>
 <![CDATA[&=\colvector{4\\0\\0\\0}+]]>
 \colvector{-3x_3\\-x_3\\x_3\\0}+
-<![CDATA[\colvector{2x_4\\3x_4\\0\\x_4}&&]]>\text{<acroref type="definition" acro="CVA" />}\\
+<![CDATA[\colvector{2x_4\\3x_4\\0\\x_4}&&]]><acroref type="definition" acro="CVA" />\\
 <![CDATA[&=\colvector{4\\0\\0\\0}+]]>
 x_3\colvector{-3\\-1\\1\\0}+
-<![CDATA[x_4\colvector{2\\3\\0\\1}&&]]>\text{<acroref type="definition" acro="CVSM" />}\\
+<![CDATA[x_4\colvector{2\\3\\0\\1}&&]]><acroref type="definition" acro="CVSM" />\\
 </alignmath></p>
 
 <p>We will develop the same linear combination a bit quicker, using three steps.  While the method above is instructive, the method below will be our preferred approach.</p>
 \colvector{ 3x_4\\ -4x_4\\ 0\\ x_4\\ 0\\ 0\\ 0 }
 +
 \colvector{ -9x_7\\ 8x_7\\ 0\\ 0\\ 6x_7\\ -7x_7\\ x_7 }
-<![CDATA[&&]]>\text{<acroref type="definition" acro="CVA" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="CVA" />\\
 <![CDATA[&=]]>
 \colvector{15\\ -10\\ 0\\ 0\\ 11\\ -21\\ 0 }
 +
 x_4\colvector{ 3\\ -4\\ 0\\ 1\\ 0\\ 0\\ 0 }
 +
 x_7\colvector{ -9\\ 8\\ 0\\ 0\\ 6\\ -7\\ 1 }
-<![CDATA[&&]]>\text{<acroref type="definition" acro="CVSM" />}
+<![CDATA[&&]]><acroref type="definition" acro="CVSM" />
 </alignmath>
 We will now develop the same linear combination a bit quicker, using three steps.  While the method above is instructive, the method below will be our preferred approach.</p>
 
 \vectorentry{\vect{w}}{3}\vect{A}_3+
 \cdots+
 \vectorentry{\vect{w}}{n}\vect{A}_n
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="SLSLC" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="SLSLC" />\\
 <![CDATA[&=]]>
 \vectorentry{\vect{w}}{1}\vect{A}_1+
 \vectorentry{\vect{w}}{2}\vect{A}_2+
 \cdots+
 \vectorentry{\vect{w}}{n}\vect{A}_n
 +\zerovector
-<![CDATA[&&]]>\text{<acroref type="property" acro="ZC" />}\\
+<![CDATA[&&]]><acroref type="property" acro="ZC" />\\
 <![CDATA[&=]]>
 \vectorentry{\vect{w}}{1}\vect{A}_1+
 \vectorentry{\vect{w}}{2}\vect{A}_2+
 \vectorentry{\vect{w}}{3}\vect{A}_3+
 \cdots+
 \vectorentry{\vect{w}}{n}\vect{A}_n
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="SLSLC" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="SLSLC" />\\
 <![CDATA[&\quad\quad]]>
 +
 \vectorentry{\vect{z}}{1}\vect{A}_1+
 <![CDATA[&=]]>
 \left(\vectorentry{\vect{w}}{1}+\vectorentry{\vect{z}}{1}\right)\vect{A}_1+
 \left(\vectorentry{\vect{w}}{2}+\vectorentry{\vect{z}}{2}\right)\vect{A}_2
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="VSPCV" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="VSPCV" />\\
 <![CDATA[&\quad\quad]]>
 \left(\vectorentry{\vect{w}}{3}+\vectorentry{\vect{z}}{3}\right)\vect{A}_3+
 \cdots+
 \vectorentry{\vect{w}+\vect{z}}{2}\vect{A}_2+
 \cdots+
 \vectorentry{\vect{w}+\vect{z}}{n}\vect{A}_n
-<![CDATA[&&]]>\text{<acroref type="definition" acro="CVA" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="CVA" />\\
 <![CDATA[&=]]>
 \vectorentry{\vect{y}}{1}\vect{A}_1+
 \vectorentry{\vect{y}}{2}\vect{A}_2+
 \vectorentry{\vect{y}}{3}\vect{A}_3+
 \cdots+
 \vectorentry{\vect{y}}{n}\vect{A}_n
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="SLSLC" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="SLSLC" />\\
 <![CDATA[&\quad\quad]]>
 -
 \left(
 <![CDATA[&=]]>
 \left(\vectorentry{\vect{y}}{1}-\vectorentry{\vect{w}}{1}\right)\vect{A}_1+
 \left(\vectorentry{\vect{y}}{2}-\vectorentry{\vect{w}}{2}\right)\vect{A}_2
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="VSPCV" />}\\
+<![CDATA[&&]]><acroref type="theorem" acro="VSPCV" />\\
 <![CDATA[&\quad\quad]]>
 +
 \left(\vectorentry{\vect{y}}{3}-\vectorentry{\vect{w}}{3}\right)\vect{A}_3+
 <![CDATA[&=]]>
 \vectorentry{\vect{y}-\vect{w}}{1}\vect{A}_1+
 \vectorentry{\vect{y}-\vect{w}}{2}\vect{A}_2
-<![CDATA[&&]]>\text{<acroref type="definition" acro="CVA" />}\\
+<![CDATA[&&]]><acroref type="definition" acro="CVA" />\\
 <![CDATA[&\quad\quad]]>
 +\vectorentry{\vect{y}-\vect{w}}{3}\vect{A}_3+
 \cdots+

File src/section-LDS.xml

 Since the $\alpha_i$ cannot all be zero, choose one, say $\alpha_t$, that is nonzero.  Then,
 <alignmath>
 \vect{u}_t
-<![CDATA[&=\frac{-1}{\alpha_t}\left(-\alpha_t\vect{u}_t\right)&&]]>\text{<acroref type="property" acro="MICN" />}\\
+<![CDATA[&=\frac{-1}{\alpha_t}\left(-\alpha_t\vect{u}_t\right)&&]]><acroref type="property" acro="MICN" />\\
 <![CDATA[&=]]>
 \frac{-1}{\alpha_t}\left(
 \alpha_1\vect{u}_1+
 \alpha_{t+1}\vect{u}_{t+1}+
 \cdots+
 \alpha_n\vect{u}_n
-<![CDATA[\right)&&]]>\text{<acroref type="theorem" acro="VSPCV" />}\\
+<![CDATA[\right)&&]]><acroref type="theorem" acro="VSPCV" />\\
 <![CDATA[&=]]>
 \frac{-\alpha_1}{\alpha_t}\vect{u}_1+
 \cdots+
 \frac{-\alpha_{t+1}}{\alpha_t}\vect{u}_{t+1}+
 \cdots+
 \frac{-\alpha_n}{\alpha_t}\vect{u}_n
-<![CDATA[&&]]>\text{<acroref type="theorem" acro="VSPCV" />}
+<![CDATA[&&]]><acroref type="theorem" acro="VSPCV" />
 </alignmath>
 </p>
 
 \beta_{t+1}\vect{u}_{t+1}+