The question is:
Determine if sets $U$ and $V$ are subspaces of vector space $\ P_3 \ (P_3 = \{a_2x^2 + a_1x +a_0: a_2, a_1, a_0 \in P_3 \}) $, when:
- $U = \{a_2x^2 + a_1x +a_0 \in P_3: a_2x^2 + a_1x +a_0 = 0 \land 2a_2 + a_1 = 0 \}$
- $V = \{p(x) = a_2x^2 + a_1x +a_0 \in P_3: p(0) < p(1)\}$
I know that $W$ is a subspace of $V$ if:
- $W \neq \emptyset $
- $\forall x, y \in W, x + y \in W $
- $\forall x \in W, \forall \alpha \in \mathbb R, \alpha x \in W $
What I am not sure about are those conditions in sets $U$ and $V$. What is the right approach when solving this type of task?