theorem for the direct sum of finite dimensional vector spaces
Proof. Suppose that . Then, by definition, and . The dimension theorem for subspaces states that
and the first direction of the claim follows.
For the other direction, suppose and . Then the dimension theorem theorem for subspaces implies that
Now is a subspace of with the same dimension as so, by Theorem 1 on this page (http://planetmath.org/VectorSubspace), . This proves the second direction.
|Title||theorem for the direct sum of finite dimensional vector spaces|
|Date of creation||2013-03-22 13:36:17|
|Last modified on||2013-03-22 13:36:17|
|Last modified by||matte (1858)|