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Let $S$ be the subspace of matrix 2x2 in $\mathbb R$ formed by symetrical matrix and $T$ the one formed by the matrix of trace zero.

Let $A=\left(\begin{array}{cc} 1 & 0\\ 2 & 0 \end{array}\right)$ and $f:S\rightarrow T$ linear given by $f(M)=AM-MA.$

So the first question is to get $\mathcal B_1$ and $\mathcal B_2$ basis os $S$ and $T$ so the matrix of $f$ is $A=\left(\begin{array}{cc} I_r & 0\\ 0 & 0 \end{array}\right)$.

So I ended up with:

$\mathcal B_1$ :$\left(\begin{array}{cc} 0& 0\\ 1 & 0 \end{array}\right)$$\left(\begin{array}{cc} 0 & 1\\ 0 & 0 \end{array}\right)$$\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right)$.

$\mathcal B_2$ : $\left(\begin{array}{cc} 0& 0\\ -1 & 0 \end{array}\right)$$\left(\begin{array}{cc} -2 & 1\\ 0 & 2 \end{array}\right)$$\left(\begin{array}{cc} 1 & 0\\ 1 & 0 \end{array}\right)$

then the matrix of $f$ is: $\left(\begin{array}{cc} 1 & 0 & 0\\ 0 & 1 &0\\0&0&0\end{array}\right)$.

But now they ask me to find the applications $g:S\rightarrow S$ with $f\circ g=f$ and express their matrix respect $\mathcal B_1$(g is linear).

So here I got that those g are the ones with matrix: $\left(\begin{array}{cc} 1 & 0 & 0\\ 0 & 1 &0\\a&b&c\end{array}\right)$ .

With $a,b,c\in \mathbb R$. But I don't know how to start with the next question. Of all those g, find the ONLY ONE that meets $Im\space g \subseteq T$.

How to start? Any hint?

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  • $\begingroup$ How are the members of your $\mathcal B_i$ elements of $S$ or $T$? $\endgroup$ – Berci Dec 31 '18 at 0:39
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Hints:

  • note that $S\cap T = \langle\left(\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}\right) \rangle$
  • $f\left(\begin{matrix}m_{11} & m_{12} \\ m_{21}& m_{22} \end{matrix}\right)=\left(\begin{matrix} 0 & -m_{12} \\ -m_{21}& 0 \end{matrix}\right)$
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  • $\begingroup$ still don't know how to follow :( $\endgroup$ – iggykimi Dec 31 '18 at 14:49

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