# Find matrix of linear map of polynomials

For the question below, I am not sure how to find the desired matrix relative to the standard basis. $$P_3$$ refers to polynomials with up to degree $$3$$

for the linear map

$$T:P_3(\mathbb{R})\rightarrow\mathbb{R^4}, p\mapsto \big(p(0), p(1), p\prime(0), p\prime(1) \big)$$

a) Find the matrix $$[T]_\beta^\gamma$$ of $$T$$ relative to the standard bases of $$P_3(\mathbb{R})$$ and of $$\mathbb{R^4}$$

b) Show that $$T$$ is an isomorphism

• Do you know the definition of a linear transformation's matrix with respect to certain basis? If so, what have you tried so far? Nov 4 '17 at 2:25

Let $B := \{1, X, X^2, X^3\}$ be the standard basis of $\mathbb{R}_{\leq3}[X]$, and $B' = \{e_1, e_2, e_3, e_4\}$ the standard basis of $\mathbb{R}^4$. Now, by definition, $[T]_{B}^{B'}$ has on column $i$ the coordinates in $B'$ of the image of the $i$-th vector of $B$. So, since $B'$ is the canonical basis, taking coordinates is redundant: it suffices to place $T(X^{i-1})$ in each column $i$. So,
$$T(1) = (1,1,0,0)\\ T(X) = (0,1,1,1)\\ T(X^2) = (0,1,0,2)\\ T(X^3) = (0,1,0,3)$$
$$[T]_{B}^{B'}= \begin{bmatrix} 1 & 0 & 0 & 0\\ 1 & 1 & 1 & 1\\ 0 & 1 & 0 & 0\\ 0 & 1 & 2 & 3 \end{bmatrix}$$
To finish we must conclude that $T$ is an isomorphism. This follows from the fact that $[T]_{B}^{B'}$ is invertible, since
$$\det \begin{bmatrix} 1 & 0 & 0 & 0\\ 1 & 1 & 1 & 1\\ 0 & 1 & 0 & 0\\ 0 & 1 & 2 & 3 \end{bmatrix} = \det \begin{bmatrix} 1 & 1 & 1\\ 1 & 0 & 0\\ 1 & 2 & 3 \end{bmatrix} = \det \begin{bmatrix} 1 & 1\\ 2 & 3 \end{bmatrix} = 1 \neq 0$$