# Adding and Subtracting Vector Subspaces

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!

• I've never seen this subtraction notation used for subspaces (though sometimes for subsets) - probably because, as you note, it doesn't really act very much like subtraction, so it's rather misleading to write it as if it were. Commented Oct 14, 2019 at 23:39

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 $$U+W \subseteq U-W$$)