Let $V$ be the space of all $n$ x $n$ matrices, define $T:V\rightarrow V$ as $T(A)=AB-BA$ where $A\in V$ and $B\in V$ is a fixed matrix. We are to show that
(a) $T$ is linear (b) $T$ is not one-one (c) $T$ is not onto
For (a), I was able to show $T(k_1A_1+k_2A_2)=k_1T(A_1)+k_2T(A_2) \forall A_1,A_2\in V$ and $k_1,k_2\in \Bbb R$. Hence $T$ is linear.
For (b), I was able to show $T(O)=T(B)=T(I_n)=O$ where $I_n$ is the $n$ x $n$ identity matrix and $O$ is the $n$ x $n$ null matrix and $I_n,O,B\in V$. Hence $T$ is not one-one.
For (c), I need help.
An endomorphism of a vector space is ono-to-one if and only if it is onto. Since you proved that it is not one-to-one…
You can also note that $\operatorname{tr}(AB-BA)=0$. Therefore, for instance, there is no $A$ such that $T(A)=\operatorname{Id}$.