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

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Also see

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

Sources

• 2015: Sheldon Axler: Linear Algebra Done Right (3rd ed.)