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
$dim(S+T)+dim(S\cap T)$  $=$  $dimS+dimT.$ 
References
 1 S. Lang, Linear Algebra, AddisonWesley, 1966.
 2 W.E. Deskins, Abstract Algebra, Dover publications, 1995.
Title  vector subspace 
Canonical name  VectorSubspace 
Date of creation  20130322 11:55:24 
Last modified on  20130322 11:55:24 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  20 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 1500 
Synonym  subspace 
Synonym  linear subspace 
Related topic  VectorSpace 
Related topic  LinearManifold 
Defines  dimension theorem for subspaces 
Defines  proper vector subspace 