Let $P\_{m} := \{p(t) = a\_{0} + a\_{1}\cdot t + \ldots + a\_{m}\cdot t^m\}$ be the set of all polynomials with degree $\leq$ $m$. 

$V := \{p(t) \in P\_{3} | p(0)=p(1)=0 \}$

now I need to show that $V$ is a linear subspace of $P\_{3}$ and find a basis for it. This is how I proceede but I'm not quite sure if I'm getting this right and I don't have a solution to this exercise to check. 

to prove that a vectorspace is a linear subspace we need to check the two conditions

$ \bullet x+y \in V$ 

$\bullet \alpha \cdot x \in V$

I proceeded as such:

$x(t),y(t) \in P\_{3} \\$

$y\_{0} + y\_{1}t+y\_{2}t^2+y\_{3}t^3 + x\_{0} + x\_{1}t+x\_{2}t^2+x\_{3}t^3$ is obviously also in $P\_{3}$. Now is this already enough or do I need to proceed by plugging in 1 and zero for t and getting something like: $\\$

$x\_{0}+y\_{0} = 0 $ and $x\_{1}+x\_{2}+x\_{3} + y\_{1}+y\_{2}+y\_{3} = 0$??. Please help me since the space $V$ kinda confuses me. I usually know how to solve this kind of problems but for some reason I can't wrap my head around this one.

Thanks a lot in advance.

edit: thank you for your help. I managed to solve this.