Let $B$ $=$ {$x^2, x, 1$} and $S$ $=$ {$x^2+x, 2x-1, x+1$} be two basis of $P_2$. Let $T$ be a linear transformation from $P_2$ to $P_2$ such that the transition matrix from $B$ to $S$ is $\begin{pmatrix}1&2&0\\ -1&3&5\\ 2&2&-2\end{pmatrix}$. Find a basis for $Im(T)$.
So I row reduced the given matrix to get its column space, which was:
$\left\{\begin{pmatrix}1\\ -1\\ 2\end{pmatrix},\:\begin{pmatrix}2\\ 3\\ 2\end{pmatrix}\right\}$
But I am not sure on how to find a basis for the image after this. Am I supposed to use the two basis initially given? If so, how?
I do see that the answer is $\left\{x^2+x-3,\:2x^2+10x-1\right\}$ but I am not entirely sure how that came about as the coefficients here are not related to the numbers in the column space.
Any help would be highly appreciated!
After getting that column space that you got (it is correct), you should deduce that a basis of $\operatorname{Im}(T)$ is$$\bigl\{1\times(x^2+x)-1\times(2x-1)+2\times(x+1),2\times(x^2+x)+3\times(2x-1)+2\times(x+1)\bigr\},$$which is equal to$$\{x^2+x+3,2x^2+10x-1\}.$$