I have been attempting quite a few of these "find the matrix from the linear transformation" problems and I have been stuck on this problem for some time. For the standard bases what I have is
\begin{bmatrix} p(1) & 0 & 0 \\ 0 & p\prime(1) & 0 \\ \\ 0 & 0 & p\prime\prime(-1) \end{bmatrix}
and here's what I have for the other basis
\begin{bmatrix} p(1) & p(1) & p(1) \\ 0 & p\prime(1) & p\prime(1) \\ \\ 0 & 0 & p\prime\prime(-1) \end{bmatrix}
Thanks in advance!
Let $T:P_3(\mathbb{R})\rightarrow\mathbb{R^3}$ be the linear transformation given by $$T(p)=(p(1), p\prime(1), p\prime\prime(-1))$$.
Find the matrix of $T$ relative to the standard basis of $P_3(\mathbb{R})$ and the basis $$v_1=(1,0,0), v_2=(1,1,0), v_3=(1,1,1)$$ of $\mathbb{R^3}$