Is S a subspace of P2? Give a proof for your answer.

Question

Let S = {p(x) ∈ P2|p(0)− p(4) = 0}. Is S a subspace of P2? Give a proof for your answer.

So I know for a subspace proof you need to prove that S is non-empty, closed under addition, and scalar multiplication.

However, does this mean that p(0) = 0 will satisfy the non-empty component? Also, will S(x+y) = (p(0)+p'(0)) - ((p(4)+p'(4)) = S(x) + S(y) satisfy the addition component? And finally, is S(kx) = (kp(0)) - ((kp(4)) = (k)(p(0) - p(4)) satisfy the multiplication component?

Mostly I'm just confused as to prove this, am I going about it the right way? Thank you!

Answer 1

What you have to do is the following:

1. Check that $0 \in S$. This is obvious since $0(0)-0(4)=0-0=0$.

2. Pick two arbitrary elements $p(x), q(x) \in S$ and check that $p(x)+q(x) \in S$. $$(p(0)+q(0)) - (p(4)+q(4)) = (p(0)-p(4)) +(q(0)-q(4)) = 0+0=0$$

3. Pick an arbitrary element $p(x) \in S$ and an arbitrary $k \in \Bbb K$ and check that $kp(x) \in S$ $$kp(0)-kp(4)= k (p(0)-p(4))=k \cdot 0=0$$

This concludes the proof.