Polynomial subspace Wondering abut this set : $E=(p(X) \ \in \mathbb{R}[X]; Xp(X)+p'(X)=0)$, is it a subspace of $\mathbb{R}[X] $?
I definitely think it is because it only includes the zero polynomial but how could we prove it ? I usually take $u$ and $v$ which are in the set and then prove that $\lambda u+v \ \in$ the set but don't see how to proceed  here.

Hint: Find all elements of the set $E$.

 Hint 2: Look at the degrees and leading coefficients.

