# theorem for the direct sum of finite dimensional vector spaces

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 (http://planetmath.org/VectorSubspace), $V=S+T$. This proves the second direction. $\Box$

Title theorem for the direct sum of finite dimensional vector spaces TheoremForTheDirectSumOfFiniteDimensionalVectorSpaces 2013-03-22 13:36:17 2013-03-22 13:36:17 matte (1858) matte (1858) 8 matte (1858) Theorem msc 15A03