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I am trying to solve this two questions:

  1. given a linear transformation: $T:M_n(\mathbb{C}) \rightarrow M_n(\mathbb{C})$, $T(A)=A-2A^T$, what is the representative matrix on this transformation with the standart base.

  2. Let $T: \mathbb{F}^2 \rightarrow \mathbb{F}^2$ a linear tranformation such that $T^2=0$, Prove that $T=0$, or there exists a basis of $M_2(\mathbb{F})$ in which the representative matrix is: $\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}$.

Thank you.

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Hint

For the first question here a very related question as Ewan Delanoy said.

For your second question assume that $T\ne0$ then there's $u\in\Bbb F^2$ such that $T(u)=v\ne0$. Prove that $\mathcal B=(v,u)$ is a basis of $\Bbb F^2$ and the matrix of $T$ in this basis has the desired form.

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