# 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 $V\neq W$, 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. 1.

Every vector space is a vector subspace of itself.

2. 2.

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

3. 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. 4.

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

5. 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

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

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

 $\displaystyle\dim(S+T)+\dim(S\cap T)$ $\displaystyle=$ $\displaystyle\dim S+\dim T.$

## References

• 1 S. Lang, Linear Algebra, Addison-Wesley, 1966.
• 2 W.E. Deskins, Abstract Algebra, Dover publications, 1995.
 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