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# 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*

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