Determine if U a subspace of $P_3$?

Question

U= The set of all polynomials in $P_3$ with constant term 0.

I'm not exactly sure how to go about this question... I know I have to prove Axioms A1, A4, and S1.

A1.If x∈W and y∈W, then x+y∈W.
A4.W is nonempty, W≠∅.
S1.If α∈C and x∈W, then αx∈W.

However, I'm not exactly sure how to do so... Thanks in advance!

Answer 1

As you mentioned you have to prove that the subspace is $a.$ not empty b. closed to linear combinations.

$U=\{a_0+a_1x+a_2x^2+a_3x^3:a_0=0\}$

So $(0,0,0,0)\in U$ so $U$ is not empty.

Let there be $u_1,u_2\in U$ and $\alpha\in \mathbb{F}$ so $u_1+\alpha\cdot u_2=a_1x+a_2x^2+a_3x^3+\alpha(b_1x+b_2x^2+b_3x^3)=a_1x+a_2x^2+a_3x^3+\alpha \cdot b_1x+\alpha \cdot b_2x^2+\alpha \cdot b_3x^3)=(a_1+\alpha \cdot b_1)x+(a_2+\alpha \cdot b_2)x^2+(a_3+\alpha \cdot b_3)x^3 \in U$

So $U$ is a subspace of $P_3$