Here is the question I am trying to answer the second part of it:\
Let $V$ be a vector space of all $n$ by $n$ matrices over the field $F,$ and let $B$ be a fixed $n$ by $n$ matrix. Let $T: V \to V$ be defined by $$T(A) := AB - BA$$ Is $T$ a linear map? Is it an isomorphism?
My question is:
I managed to prove that it is a linear map. But how to think if it is an isomorphism or no, I am guessing yes, could someone help me in this please?
If $B=0$ the maping $T$ equals the zero homomorphism which is not an isomorphism. Otherwise $T(B)=0$ and $B\not=0$. Thus $T$ is not injective, implying that $T$ is not an isomorphism also in this case.