I am learning linear transformations, and I understand transformation of R vector space, but I wont understand transformations of polynomials,could you help,I am trying to solve this question:
$T:P_2\to P_2$
$\begin{align} T(p1)&=5x^2+2x-9\\ T(p2)&=-5x^2+x\\ T(p3)&=-2x^2+x-4 \end{align}$
$\begin{align} p1&=-4x^2-x-4\\ p2&=-3x^2-4x-5\\ p3&=-x^2+x-5 \end{align}$
I need to find the basis of the kernel and the image of the transformation
Make sure you understand that the definition of $T$ just on $p_1,p_2,p_3$ makes sense because they form a basis of $P_2$. This is easy to check by writing $p_1,p_2,p_3$ with respect to the canonical basis $1,x,x^2$ and checking that the determinant is not zero.
To compute the kernel and the image of $T$, row-reduce the matrix of $T$ with respect to the basis $p_1,p_2,p_3$.