## Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation

## Problem 330

Let $V$ be the vector space of all $n\times n$ real matrices.

Let us fix a matrix $A\in V$.

Define a map $T: V\to V$ by

\[ T(X)=AX-XA\]
for each $X\in V$.

**(a)** Prove that $T:V\to V$ is a linear transformation.

**(b)** Let $B$ be a basis of $V$. Let $P$ be the matrix representation of $T$ with respect to $B$. Find the determinant of $P$.