Let $\ f_k:M_2(\Bbb R)\to\Bbb R[x]_{\le2}, g_k:M_2(\Bbb R)\to\Bbb R^3$ be linear maps such that $f_k\left(\begin{bmatrix} a& b \\ c&d \end{bmatrix}\right)=ax^2+kbx+d,\ g_k\left(\begin{bmatrix} a& b \\ c&d \end{bmatrix}\right)=\begin{bmatrix} a-kb \\ 0 \\ k(k-2)(b-c)+d \end{bmatrix}.$ I'm supposed to find a basis of the kernels and the images of both maps above.
Now, I understand that we should use the fact that $R[x]_{\le2}$ is isomorphic to $\Bbb R^3$, but I'm lost when it comes to the representative matrices. How can it be that multiplied by a 2x2 matrix, they give a 3x1 one?
I think they're not necessary for the kernels, but they should be for the images...
Guide:
View $M_2(\mathbb{R})$ as $\mathbb{R}^4$. A matrix is just a matter of representation.
For example find the kernel and image for the following transformation:
$$\hat{f}_k\left(\begin{bmatrix} a\\ b \\ c\\d \end{bmatrix}\right)=ax^2+kbx+d$$
After you find the conditions for the vector, reshape them back to a matrix.