Let $B={\{x^2, x, 1}\}$ and $S= {\{x^2+x, 2x-1, x+1}\}$ be two basis of $P_2$. Let $i_b$ and $i_s$ be the coordinates maps induced on $P_2$ by these two basis. Let$T: P_2\to P_2$ be a liner transformation such that
$[T]_{B,S}$=$\begin{bmatrix}1 & 2 &0 \\-1&3&5\\2 & 2&-2 \end{bmatrix}$
a) Prove that $ker(T)$=$i_b^{-1}(Null([T]_{B,S}))$ and $i_s(Im(T))=Col([T])_{B,S})$.
I know that $[T(x)]_B=[T]_{B,S}[x]_B$
but what is $i_B$ and what is its $Im$??
I am stuck with finding the $ker(T)$, I would be appreciate if someone can walk me through this problem.