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:
 * $U \cap W = 0$

Let $U + W$ be a direct sum.

Let $\mathbf v \in U \cap W$ be an arbitrary vector in $U \cap W$.

Then:
 * $\mathbf 0 = \mathbf v + \paren {-\mathbf v}$

where:
 * $\mathbf v \in U$
 * $-\mathbf v \in W$

Also see

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