vector subspace

Definition Let $V$ be a vector space over a field $F$, and let $W$ be a subset of $V$. If $W$ is itself a vector space, then $W$ is said to be a vector subspace of $V$. If in addtition $V\ne W$, then $W$ is a proper vector subspace of $V$.

If $W$ is a nonempty subset of $V$, then a necessary and sufficient condition for $W$ to be a subspace is that $a+\gamma b\in W$ for all $a,b\in W$ and all $\gamma \in F$.

0.0.1 Examples

1. Every vector space is a vector subspace of itself.

2. In every vector space, $\{0\}$ is a vector subspace.

3. If $S$ and $T$ are vector subspaces of a vector space $V$, then the vector sum

   $$S+T=\{s+t\in V\mid s\in S,t\in T\}$$

   and the intersection

   $$S\cap T=\{u\in V\mid u\in S,u\in T\}$$

   are vector subspaces of $V$.

4. Suppose $S$ and $T$ are vector spaces, and suppose $L$ is a linear mapping $L:S\to T$. Then $\mathrm{Im}L$ is a vector subspace of $T$, and $\mathrm{Ker}L$ is a vector subspace of $S$.

5. If $V$ is an inner product space, then the orthogonal complement of any subset of $V$ is a vector subspace of $V$.

0.0.2 Results for vector subspaces

Theorem 1 [1] Let $V$ be a finite dimensional vector space. If $W$ is a vector subspace of $V$ and $dimW=dimV$, then $W=V$.

Theorem 2 [2] (Dimension theorem for subspaces) Let $V$ be a vector space with subspaces $S$ and $T$. Then

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

Title	vector subspace
Canonical name	VectorSubspace
Date of creation	2013-03-22 11:55:24
Last modified on	2013-03-22 11:55:24
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	20
Author	yark (2760)
Entry type	Definition
Classification	msc 15-00
Synonym	subspace
Synonym	linear subspace
Related topic	VectorSpace
Related topic	LinearManifold
Defines	dimension theorem for subspaces
Defines	proper vector subspace