# 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?

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

• note that $$S\cap T = \langle\left(\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}\right) \rangle$$
• $$f\left(\begin{matrix}m_{11} & m_{12} \\ m_{21}& m_{22} \end{matrix}\right)=\left(\begin{matrix} 0 & -m_{12} \\ -m_{21}& 0 \end{matrix}\right)$$