Chapter 6.2, Problem 67E

Guided Proof Let $B$ be an invertible $n\times n$ matrix. Prove that the linear transformation $T:M_{n,n}\to M_{n,n}$ represented by $T\left(A\right)=AB$ is an isomorphism.

Getting started: To show that the linear transformation is an isomorphism, you need to show that $T$ is both onto and one-to-one.

(i) $T$ is a linear transformation with vector spaces of equal dimension, so by Theorem $6.8$, you only need to show that $T$ is one-to-one.

(ii) To show that $T$ is one-to-one, you need to determine the kernel of $T$ and show that it is $\left\{0\right\}$ (Theorem $6.6$). Use the fact that $B$ is an invertible $n\times n$ matrix and that $T\left(A\right)=AB$.

(iii) Conclude that $T$ is an isomorphism.