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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!

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  • $\begingroup$ 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. $\endgroup$
    – Milo Brandt
    Commented Oct 14, 2019 at 23:39

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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 $U+W \subseteq U-W$)

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