$\mathcal T (A)=B^{-1}AB$. Prove that $\mathcal T$ is an isomorphism.

Question

$B$ is an $n\times n$ invertible matrix.

Define $\mathcal T :M_{n\times n}(\mathbf F)\to M_{n\times n}(\mathbf F)$ by

$\mathcal T (A)=B^{-1}AB$. Prove that $\mathcal T$ is an isomorphism.

I think we need to check $\mathcal T$ is $1-1$ and onto.

Answer 1

To prove that $\mathcal T$ is $1-1$:

Take $X, Y\in$, and assume that $\mathcal T(X)=\mathcal T(Y)$. From that, prove that $X=Y$.

To prove that $\mathcal T$ is onto:

Take $X\in M_{n\times n}(\mathbf F)$ and find some $A\in M_{n\times n}(\mathbf F)$ such that $\mathcal T(A)=X$.

Answer 2

Since it is a linear map between finite-dimensional vector spaces with the same dimension, all you need to prove is that $\mathcal T$ is injective or surjective. But it is injective:\begin{align}\mathcal T(A)=0&\iff B^{-1}AB=0\\&\iff A=B.0.B^{-1}\\&\iff A=0.\end{align}