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 $\gamma \in F$ .

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 $V$ .

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 \begin{eqnarray*} \dim (S+T) + \dim (S\cap T) &=& \dim S + \dim T. \end{eqnarray*}

Bibliography

1

S. Lang, Linear Algebra, Addison-Wesley, 1966.

2

W.E. Deskins, Abstract Algebra, Dover publications, 1995.