# 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