## You are here

Homevector subspace

## Primary tabs

# 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. 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 $L\colon S\to T$. Then $\operatorname{Im}L$ is a vector subspace of $T$, and $\operatorname{Ker}L$ is a vector subspace of $S$.

5. 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 $\dim W=\dim V$, 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.

## Mathematics Subject Classification

15-00*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth