# HG changeset patch
# User Rob Beezer
# Date 1354512791 28800
# Node ID f461d49621f52b2b37ccaf4faffd64f880669bb0
# Parent 47911627bdde45a82dd14506aec592a407d4d524
Solution B.T50, missing implication arrows
diff --git a/changes.txt b/changes.txt
--- a/changes.txt
+++ b/changes.txt
@@ -67,6 +67,7 @@
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
~~~~~~~~~~~~~~~~~~~~~~~~~~
diff --git a/src/section-B.xml b/src/section-B.xml
--- a/src/section-B.xml
+++ b/src/section-B.xml
@@ -1121,7 +1121,7 @@
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 , .)
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, .

- 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 $C$,
+ 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 $C$,
\zerovector
@@ -1184,7 +1184,7 @@
\text{}
So we can write an arbitrary vector of $\complex{n}$ as a linear combination of the elements of $C$. In other words, $C$ spans $\complex{n}$ (). By , the set $C$ is a basis for $\complex{n}$.

- 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
+ 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