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 $U \cap W = 0$.

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 $\mathbf v \in U \cap W$, then $\mathbf 0 = \mathbf v + \paren {- \mathbf v}$, where $\mathbf v \in U$ and $- \mathbf v \in W$.

Also see

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