how can i find a linear transformation with a given condition?

Question

how can I find a linear transformation with this condition (If there is one):

$T: M_2(\mathbb{R})\to \mathbb{R}_2[x]$

$$\ker T=<\begin{pmatrix} 1 & 0 \\ -1 & 0 \\ \end{pmatrix}>$$

and how can I find the formula?

Answer 1

$M_2(\mathbb R)$ has dimension $4$ and $\mathbb R_2[x]$ has dimension $3$. So there is always a kernel. There are many possibilities for such $T$.

Let's consider $f\left(\begin{pmatrix}a&b\\c&d\end{pmatrix}\right) = a+bX+cX^2$. This does not fullfill your conditions. But can you calculate $\ker(f)$? Maybe then you have an idea how to come up with $T$.

If not: $M_2(\mathbb R)$ is not very different from $\mathbb R^4$ and $\mathbb R_2[x]$ is not very different from $\mathbb R^3$. How would you search a map $T'\colon \mathbb R^4 \to \mathbb R^3$ with $\ker(f) = \left\langle \begin{pmatrix}1\\0\\-1\\0\end{pmatrix}\right\rangle$?