vector subspace

DefinitionLet $V$ 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 subspaceis 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 linearmapping $L:S\to T$.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 withsubspaces $S$ and $T$. Then

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