# Commits

committed e07dee4

LaTeX formatting fixup

# src/section-CRS.xml

 <p>Let's determine a compact expression for the entire column space of the coefficient matrix of the system of equations that is <acroref type="archetype" acro="D" />.  Notice that in <acroref type="example" acro="CSMCS" /> we were only determining if individual vectors were in the column space or not, now we are describing the entire column space.</p>

 <p>To start with the application of <acroref type="theorem" acro="BCS" />, call the coefficient matrix $A$ and row-reduce it to reduced row-echelon form $B$,
-<alignmath>
-A&amp;=<archetypepart acro="D" part="purematrix" />
-&amp;
-B&amp;=<archetypepart acro="D" part="matrixreduced" /></alignmath>
+<alignmath>A&amp;=<archetypepart acro="D" part="purematrix" />&amp;B&amp;=<archetypepart acro="D" part="matrixreduced" /></alignmath>
 </p>

 <p>Since columns 1 and 2 are pivot columns, $D=\{1,\,2\}$.  To construct a set that spans $\csp{A}$, just grab the columns of $A$ with indices in $D$, so
 <indexlocation index="Archetype A!column space" />
 <p>The coefficient matrix in <acroref type="archetype" acro="A" /> is $A$, which row-reduces to $B$,

-<alignmath>A&amp;=<archetypepart acro="A" part="purematrix" />
-&amp;
-B&amp;=<archetypepart acro="A" part="matrixreduced" /></alignmath></p>
+<alignmath>A&amp;=<archetypepart acro="A" part="purematrix" />&amp;B&amp;=<archetypepart acro="A" part="matrixreduced" /></alignmath></p>

 <p>Columns 1 and 2 are pivot columns, so by <acroref type="theorem" acro="BCS" /> we can write
 <equation>

 <exercise type="C" number="20" rough="Repeated calculations of C(A) and R(A);  lead to rank">
 <problem contributor="chrisblack">For each matrix below, find a set of linearly independent vectors $X$ so that $\spn{X}$ equals the clolumn space of the matrix, and a set of linearly independent vectors $Y$ so that $\spn{Y}$ equals the row space of the matrix.
-<alignmath>A&amp;=<![CDATA[\begin{bmatrix} 1 & 2 & 3 & 1 \\ 0 & 1 & 1 & 2\\ 1 & -1 & 2 & 3 \\ 1 & 1 & 2 & -1 \end{bmatrix}]]>
-&amp;
-B&amp;=<![CDATA[\begin{bmatrix}  1 & 2 & 1 & 1 & 1 \\ 3 & 2 & -1 & 4 & 5\\ 0 & 1 & 1 & 1 & 2 \end{bmatrix}]]>
-&amp;
-C&amp;=<![CDATA[\begin{bmatrix}  2 & 1 & 0 \\ 3 & 0 & 3\\ 1 & 2 & -3 \\ 1 & 1 & -1 \\ 1 & 1 & -1\end{bmatrix}]]></alignmath>
+<alignmath>A&amp;=<![CDATA[\begin{bmatrix} 1 & 2 & 3 & 1 \\ 0 & 1 & 1 & 2\\ 1 & -1 & 2 & 3 \\ 1 & 1 & 2 & -1 \end{bmatrix}]]>&amp;B&amp;=<![CDATA[\begin{bmatrix}  1 & 2 & 1 & 1 & 1 \\ 3 & 2 & -1 & 4 & 5\\ 0 & 1 & 1 & 1 & 2 \end{bmatrix}]]>&amp;C&amp;=<![CDATA[\begin{bmatrix}  2 & 1 & 0 \\ 3 & 0 & 3\\ 1 & 2 & -3 \\ 1 & 1 & -1 \\ 1 & 1 & -1\end{bmatrix}]]></alignmath>
 From your results for these three matrices, can you formulate a conjecture about the sets $X$ and $Y$?
 </problem>
 </exercise>
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