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rbeezer committed f461d49

Solution B.T50, missing implication arrows

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 Typo:  LaTeX/XML formatting, \lt in Theorem VRI proof (Stephen Gower)
 Typo:  Implication arrow in wrong direction, Theorem IMR (Dan Messenger)
 Typo:  Example LIC, second sentence, in/dependence (Aidan Meacham)
+Typo:  Solution B.T50, missing implication arrows
 
 MAJOR SOURCE FORMAT CHANGE
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src/section-B.xml

 More precisely, suppose that $A$ is a square matrix of size $n$ and $B=\set{\vectorlist{x}{n}}$ is a basis of $\complex{n}$.  Prove that $A$ is nonsingular if and only if $C=\set{A\vect{x}_1,\,A\vect{x}_2,\,A\vect{x}_3,\,\dots,\,A\vect{x}_n}$ is a basis of $\complex{n}$.  (See also <acroref type="exercise" acro="PD.T33" />, <acroref type="exercise" acro="MR.T20" />.)
 </problem>
 <solution contributor="robertbeezer">Our first proof relies mostly on definitions of linear independence and spanning, which is a good exercise.  The second proof is shorter and turns on a technical result from our work with matrix inverses, <acroref type="theorem" acro="NPNT" />.<br /><br />
-<forwardimply />  Assume that $A$ is nonsingular and prove that $C$ is a basis of $\complex{n}$.  First show that $C$ is linearly independent.  Work on a relation of linear dependence on $C$,
+<implyforward />  Assume that $A$ is nonsingular and prove that $C$ is a basis of $\complex{n}$.  First show that $C$ is linearly independent.  Work on a relation of linear dependence on $C$,
 <alignmath>
 \zerovector
 <![CDATA[&=]]>
 <![CDATA[&&]]>\text{<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 />
-<reverseimply /> 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
+<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>
 <![CDATA[\vect{x}&=\lincombo{a}{x}{n}]]>
 </alignmath>