Problem 4 from https://math.berkeley.edu/~ogus/Math_54-07/Exams/midsol1.pdf
$\beta$ is a basis of $P_3$, the set of all polynomials of at most degree 3.$\beta = (x^0,x^1,x^2,x^3)$. Let $T$ be a linear transformation from $P_3$ to $R^2 = (p(-1),p(1))^T$.
(a) Find a basis for the kernel of $T$. (I know the answer, just not sure how it was gotten.)
(b) Find the matrix $T'$ basis $\beta$ and the standard basis for $R^2$.
A secondary question: (c) How do you find the matrix $T'$ basis $\beta = (x^3,x^2,x^1,x^0)$ and the standard basis (1,2)^T and (3,7) for $R^2$