theorem for the direct sum of finite dimensional vector spaces

TheoremLet $S$ 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 $dimV=dimS+dimT$ 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

$$dim(S+T)+dimS\cap T=dimS+dimT.$$

Since the dimension of the zero vector space $\{0\}$is zero, we have that

$$dimV=dimS+dimT,$$

and the first direction of the claim follows.

For the other direction, suppose $dimV=dimS+dimT$and $S\cap T=\{0\}$. Then thedimension theorem theorem for subspaces implies that

$$dim(S+T)=dimV.$$

Now $S+T$ is a subspace of $V$ with the same dimensionas $V$ so,by Theorem 1 on this page (http://planetmath.org/VectorSubspace),$V=S+T$. This proves the second direction. $\mathrm{\square }$