$\newcommand{\Ker}{\mathrm{Ker}}$Find a basis for the vector space $V$ so that the first $\dim(\Ker(T))$ vectors are a basis for $\Ker(T)$, $T:V\rightarrow W $ a linear transformation.
$T:R^4 \rightarrow P_2(R)$ defined by $T(a_1, ..., a_4)=(a_1+a_2)+(a_2+a_3)x+(a_3+a_4)x^2$
$T:P_n(R) \rightarrow P_n(R)$, which is given by differentiation.
In terms of 1), does the matrix look something like $\begin{bmatrix}a_1+a_2&0&0\\0&a_2+a_3&0\\0&0&a_3+a_4\end{bmatrix}$? The answer says that $\{(1,-1,1,-1), e_1, e_2, e_3\}$ is the basis, but why?
For 2), the answer says $\{1, x, ..., x^n\}$ is the basis, but I still cannot write out the matrix.