Theorem Let $S$ and $T$ be subspaces of a finite dimensional vector space $V$ . Then $V$ is the direct sum of $S$ and $T$ , i.e., $V=S\oplus T$ , if and only if $\dim V = \dim S + \dim T$ and $S\cap T = \{0\}$ .

Proof. Suppose that $V=S\oplus T$ . Then, by definition, $V=S+T$ and $S\cap T=\{0\}$ . The dimension theorem for subspaces states that $$\dim (S+T) + \dim S\cap T = \dim S + \dim T.$$ Since the dimension of the zero vector space $\{0\}$ is zero, we have that $$\dim V = \dim S + \dim T,$$ and the first direction of the claim follows.

For the other direction, suppose $\dim V = \dim S + \dim T$ and $S\cap T = \{0\}$ . Then the dimension theorem theorem for subspaces implies that $$\dim (S+T) = \dim V.$$ Now $S+T$ is a subspace of $V$ with the same dimension as $V$ so, by Theorem 1 on this page, $V=S+T$ . This proves the second direction.