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vector subspace (Definition)

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
    $\displaystyle S+T=\{s+t \in V \mid s\in S, t\in T\} $
    and the intersection
    $\displaystyle 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

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

Bibliography

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



"vector subspace" is owned by yark. [ full author list (3) | owner history (2) ]
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See Also: vector space, linear manifold

Other names:  subspace, linear subspace
Also defines:  dimension theorem for subspaces, proper vector subspace
Keywords:  vector spaces

Attachments:
proof of the dimension theorem for subspaces (Proof) by yark
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Cross-references: finite dimensional, orthogonal complement, inner product space, linear mapping, intersection, sum, vector, necessary and sufficient, subset, field, vector space
There are 153 references to this entry.

This is version 15 of vector subspace, born on 2001-10-29, modified 2007-02-28.
Object id is 624, canonical name is VectorSubspace.
Accessed 33516 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
Pending Errata and Addenda
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