I am working on this practice problem, and I was wondering if I could get some help.
I have a $T$:$\mathbb{R^{2x2}}\to \mathbf{P_{2}}$, that is, from 2x2 matrices to polynomials of degree at most 2. The transformation is given as following: $T\left(\begin{bmatrix} a & b\\ c & d \end{bmatrix}\right) = a-c+2d+(b+2c-d)t+(a-c+3d)t^{2}.$
To get the basis of kernel of $T$, I solved a system of equations needed to get the 'O' element in the $\mathbf{P_{2}}$ -- $a-c+2d=0$, $b+2c-d=0$ and $a-c+3d=0$. As a result, I got the basis of the kernel equal to $\begin{bmatrix} 1 & -2\\ 1 & 0\\ \end{bmatrix}.$
When it comes to image, if I understand correctly, I need to factor out all the variables separately, to see what is it that they span. So I got $a(1+t^{2})+b(t)+c(-t^{2}+2t-1)+d(3t^{2}-t+2)$. So would I be correct in saying that these three polynomials (without the coefficients $a$, $b$, $c$, and $d$) form the basis for the image $T$? Thank you!