Internal Direct Sums

Def: U+W

If $V$ is a vector space with subspaces $U$ and $W$ then $U+W=\{u+w:u\in U,w\in W\}$

$U+W=\text{span}(U\cup V)$

Def: Internal Direct Sum

If $V$ is a vector space and $U,W\in V$ are subspaces then $V$ is the internal direct sum of $U\text{ and }V$written as $V=U\oplus W$ if:
1. $V=U+W$
2. $U\cap W=\{0_V\}$

Thm: Equivalencies

Suppose $V$ is a vector space with subspaces $U\text{ and }W$. The following are equivalent:

1. $V=U\oplus W$
2. Every $v\in V$ can be written uniquely as $v=u+w$ for some $u\in U,v\in V$
3. If $B_u,B_w$ are disjoint bases for $U\text{ and }W$ then $B_u\bigsqcup B_w$ is a basis for the whole space