# Intersection Condition for Direct Sum of Subspaces

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## 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