# Find subspaces $W$ and $Y$ of $\mathbb{R}^3$ having the property that $W \cup Y$ is not a subspace of $\mathbb{R}^3$.

I'm prepping myself for graduate linear algebra this fall by attempting self-teach myself some of the "basics" of fields, vectors, etc. found in such linear algebra course. I really don't understand how to attempt this question. Any starting point would be greatly appreciated!

• As an exercise, one can actually prove that $W \cup Y$ is never a subspace unless $W \subseteq Y$ or vice-versa. So just choose your favorite subspaces that don't contain the other. – user61527 Jun 10 '14 at 2:44
• As pointed out by user61527, "most" choices of $W$ and $Y$ will be examples. But one might as well use geometric intuition. One-dimensional subspaces are lines through the origin, two-dimensional subspaces are planes through the origin. – André Nicolas Jun 10 '14 at 2:49
• It might be easier to look for an example in $\mathbb{R}^2$? – copper.hat Jun 10 '14 at 5:05

Hint: There are three properties of subspaces. If $W$ and $Y$ are two subspaces of $\mathbb{R}^3$, two of the three properties are satisfied by $W\cup Y$. Find the one which isn't automatically satisfied and then design an example based on that.