For the question below, would it be enough to show that the null space of $T$ is equal to $0$ for part b) and how could I show this? I am also having trouble finding the matrix for part a). $P_3$ refers to to the polynomial 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
You could apply $T $ to standard basis elements for $P_3$ and see what you get... These should be the columns of your matrix. ..
For the second part, it looks pretty straightforward that $T(p)=0 \implies p=0$ because $p (0)=a_0$ and $p'(0)=a_1$. Also, $p(1)=a_3+a_2+a_1 $ and $p'(1)=3a_3+2a_2$. It is easy from this that all $a_i =0$. (Of course $p(x)=a_3x^3+a_2x^2+a_1x+a_0$ here )