Suppose that $U$ is a subspace of $V$. What is $U+U$? Suppose that $U$ is a subspace of $V$. What is $U+U$?
Why does $U+U = U?$ I want to think of this geometrically, say in $\mathbb{R^{3}}$  we have some random plane in space that intersects the origin. How is this subspace, when added to itself, equal to itself?

$U+U$ is the subset of $V$ consisting of everything that you can possibly get when you add two elements of $U$ to each other. Since $U$ is a subspace, adding any two of its elements lands you back in the subspace, so you get $U+U \subseteq U$. Conversely, $u=u+0$ for all $u \in U$, and so $U \subseteq U+U$.
In general, given subspaces $U, W \le V$, the sum $U+W$ is what you can possibly obtain by adding a vector in $U$ and a vector in $W$. Geometrically thinking, you can imagine dragging $U$ along $W$, then $U+W$ is the space it fills up.
For instance, if $\ell_1$ and $\ell_2$ are two different lines through the origin in $\mathbb{R}^3$ then $\ell_1 + \ell_2$ is the (unique) plane containing both lines. Why? Because if you drag one line along another line, the space it sweeps out is a plane.
