TITLE: Adding and Subtracting Vector Subspaces
QUESTION [5 upvotes]: Let $U$ and $W$ be two vector subspaces of some other arbitrary vector space. We define $U - W = \left\{ \vec{u} - \vec{w} \, \big| \, \vec{u} \in U, \vec{w} \in W \right\}$. We similarly define $U + W = \left\{ \vec{u} + \vec{w} \, \big| \, \vec{u} \in U, \vec{w} \in W \right\}$. My question is:
Why is $U - U = \left\{ 0 \right\}$ false? From my understanding, subtracting something from itself always results in zero, no?
Similarly, why is $U - W = U + W$ true?
Thanks for the clarification!

REPLY [3 votes]: For an example, let's define $A = \{1,2,3,4,5\}$ and $A-A = \{a-b|a,b\in A\}$. There are many elements of $A-A$ that are nonzero. Can you see why the same goes for vectors?
As for $U-W=U+W$, since $U$ and $W$ are subspaces of a vector space, we have $u \in U \implies cu \in U$ for a constant $c$ and similarly for $W$. So $u-w= u+ (-w)$, or $u+w = u-(-w)$. (Essentially show that $U-W \subseteq U+W$ and $U+W \subseteq U-W$)