Let $V$ be the real vector space of 2x2 matrices and $End (V)$ the space of all linear transformations of V in V. $$T: V \rightarrow End (V)$$
$$T(A)(B)=AB-BA$$
I have to prove that this is a linear transformation, to characterize the subspace of the matrices $A$: $T(A)=0$, to show that if $A^2=0$ then $T(A)^3=0$ and other things, but the problem is that I don´t understand the transformation, I mean, if you could give me an example of this, it would be great. I can´t see how this transformation goes from the space of $2$x$2$ matrices to the space of all linear transformations.
PS: I´m sorry if the question is unclear, I am Spanish so if you do not understand something, please ask me.