Difference between sum and direct sum

Question

What is the difference between sum of two vectors and direct sum of two vector subspaces?

My textbook is confusing about it. Any help would be appreciated.

Answer 1

Direct sum is a term for subspaces, while sum is defined for vectors. We can take the sum of subspaces, but then their intersection need not be $\{0\}$.

Example: Let $u=(0,1),v=(1,0),w=(1,0)$. Then

• $u+v=(1,1)$ (sum of vectors),
• $\operatorname{span}(v)+\operatorname{span}(w)=\operatorname{span}(v)$, so the sum is not direct,
• $\operatorname{span}(u)\oplus\operatorname{span}(v)=\Bbb R^2$, here the sum is direct because $\operatorname{span}(u)\cap\operatorname{span}(v)=\{0\}$,
• $u\oplus v $ makes no sense in this context.

Note that the direct sum of subspaces of a vector space is not the same thing as the direct sum of some vector spaces.

Answer 2

In Axler's Linear Algebra Done Right, he defines the sum of subspaces $U + V$ as

$\{u + v : u \in U, v \in V \}$.

He then says that $W = U \oplus V$ if

(1) $W = U + V$, and

(2) The representation of each $w$ as $u + v$ is unique.

This is a different way of presenting these definitions than most texts, but it's equivalent to other definitions of direct sum.

In anyone's book, the sum and direct sum of subspaces are always defined; and the sum of vectors is always defined; but there's no such thing as a direct sum of vectors.