theorem for the direct sum of finite dimensional vector spaces


Theorem Let S and T be subspacesPlanetmathPlanetmath of a finite dimensional vector spaceMathworldPlanetmath V. Then V is the direct sumPlanetmathPlanetmathPlanetmath of S and T, i.e., V=ST, if and only if dimV=dimS+dimT and ST={0}.

Proof. Suppose that V=ST. Then, by definition, V=S+T and ST={0}. The dimension theorem for subspaces states that

dim(S+T)+dimST=dimS+dimT.

Since the dimensionPlanetmathPlanetmath 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 ST={0}. Then the dimension theorem theorem for subspaces implies that

dim(S+T)=dimV.

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.

Title theorem for the direct sum of finite dimensional vector spaces
Canonical name TheoremForTheDirectSumOfFiniteDimensionalVectorSpaces
Date of creation 2013-03-22 13:36:17
Last modified on 2013-03-22 13:36:17
Owner matte (1858)
Last modified by matte (1858)
Numerical id 8
Author matte (1858)
Entry type Theorem
Classification msc 15A03