Subspaces questions

Question

Let $ W = \{(x,x,xy,y,y): x,y \in R\}$. Is W a subspace of $R^5$? Is W contained in a proper subspace? Does W contain a subspace?

No because $(-1,-1,-1,-1,-1)$ cannot be obtained. I think so but can't find one. No beacuase it is not even a subspace.

Answer 1

Your first reasoning is correct, as $v =(1,1,1,1,1)\in W$, but $-v\not \in W $, so $W $ cannot be a subspace of $\mathbb{R}^5$.

Clearly $W \subseteq \mathbb{R}^5$ which is a subspace of itself. Regarding whether $W $ is contained in a proper subspace: Let $V$ be the set of all vectors $(x,x,z,y,y) \in \mathbb{R}^5$ such that $x,y,z \in \mathbb {R}$. You can verify yourself that V is a subspace containing $W$.

Finally, the subspace containing just the zero vector is a subset of $W $.