Intersection Condition for Direct Sum of Subspaces

Theorem
Let $U$ and $W$ be subspaces of a vector space $V$. Then $U+W$ is a direct sum if and only if $U\cap W = 0$.

Proof
We must first prove that if $U+W$ is a direct sum, then $U\cap W = 0$

Suppose $U+W$ is a direct sum. If a vector $v \in U\cap W$, then $0 = v + (-v)$, where $v \in U$ and $-v \in W$.

Also see

 * Two-Step Vector Subspace Test
 * Null Space is Subspace