Prove the function $p \mapsto T(p) = (p(1) , p(3))$ is a linear transformation

Question

Let $P_n$ be the vector space of polynomials with degree at most $n$. Prove that the function $T : P_3 \to \Bbb R_2$ defined by $$ T(p) = (p(1) , p(3)) $$ is a linear transformation

Answer 1

Here are a whole lot of questions where the answer to one should come naturally from the answer to the one before. Learn these questions by heart. They are the questions you need to ask yourself every time you're asked to show that a function is a linear map.

Say you have two polynomials $p, q$. What is $T(p+q)$? What is $T(p) + T(q)$? Are those equal as vectors in $\Bbb R^2$?

What polynomial corresponds to the $0$ element of the vector space $P_3$? What vector in $\Bbb R^2$ do you get when you apply $T$ to that polynomial?

Given a polynomial $p$ and a real number $a$, what polynomial is $ap$? What is $T(ap)$? What is $a\cdot T(p)$? Are those two equal as vectors in $\Bbb R^2$?