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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.

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$T$ sends a matrix $A$ to an endomorphism, so we denote $T$ as $$T\colon\begin{cases}V\to\mathrm{End}(V)&\\ A\mapsto T(A)\colon\begin{cases}V\to V&\\ B\mapsto T(A)(B):=AB-BA. \end{cases} \end{cases}$$

So for example $T$ sends $I_2$ to $$T(I_2)\colon\begin{cases}V\to V&\\ B\mapsto T(I_2)(B)=I_2B-BI_2=0.\end{cases}$$ Claryfied?

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  • $\begingroup$ A little bit, thanks, one question though, is B a fixed matrix? or variable? I mean, if they ask me to look for the null space (kernel), I should look for a condition in $A$ so $T(A)=0$ for all B, right? $\endgroup$
    – Lessa121
    Dec 15 '13 at 18:37
  • $\begingroup$ $B$ is arbitrary. So $A$ belongs to the kernel of $T$ iff $AB=BA$ for all $B$, as far as I can see. $\endgroup$ Dec 16 '13 at 10:38
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Denote by $sl(V)$ the linear transformations of $V$ with zero trace (as matrices). It is obvious that $T(End(V))\subset sl(V)$. However, it is possible to show by explicit calculations with basis vectors that we have equality here: $T(End(V))= sl(V)$. This answers the second question.

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