How to define this application g to meet $Im \space g \subseteq T$? 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?

Hints: 


*

*note that $S\cap T = \langle\left(\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}\right) \rangle$

*$f\left(\begin{matrix}m_{11} & m_{12} \\ m_{21}& m_{22} \end{matrix}\right)=\left(\begin{matrix} 0 & -m_{12} \\ -m_{21}& 0 \end{matrix}\right)$
