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Vector space

Revision as of 14:14, 4 November 2006 by Bubka (talk | contribs)

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A vector space over a field (frequently the real numbers) is an object which arises in linear algebra and abstract algebra. A vector space $V$ over a field $F$ consists of a set (of vectors) and two operations, vector addition and scalar multiplication, which obey the following rules:

Axioms of a vector space

  • Scalar multiplication is associative, so if $r, s \in F$ and ${\mathbf v} \in V$ then $(rs){\mathbf v} = r(s{\mathbf v})$.
  • Scalar multiplication is distributive over both vector and scalar addition, so if $r \in F$ and ${\mathbf v, w} \in V$ then $r({\mathbf v + w}) = r{\mathbf v} + r{\mathbf w}$.
  • if $x \in V$, $1.{\mathbf x}={\mathbf x}$

Subspaces

If $S \subseteq V$, and $S$ is a vector space itself, then it is called a subspace of $V$.

Independent Subsets

Let $V$ be any vector space. Let $I$ be a subset of $V$ such that no linear combination of elements of $I$ with coefficients not all zero gives the null vector. Then $I$ is said to be a linearly independent subset of $V$. An independent subset is said to be maximal if on adding any other element it ceases to be independent.

Linear Manifolds

Let $X$ be a subset of some vector space $V$. Then it can be proved that the set of all linear combinations of the elements of $X$ forms a vector space. This space is said to have been generated by $X$, and is called the linear manifold $M(X)$ of $X$.

Generating Subset

If $X$ is a subset of a vector space $V$, such that $M(X) = V$, $X$ is said to be a generating subset of $V$. A generating subset is said to be minimal if on removing any element it ceases to be generating.

Basis and dimension

The following statements can be proved using the above definitions:

  • All minimal generating subsets have the same cardinality.
  • All maximal independent subsets have the same cardinality.
  • The cardinality of an independent subset can never exceed that of a generating subset.

An independent generating subset of $V$ is said to be its basis. A basis is always a maximal independent subset and a minimal generating subset. As can be easily seen, the cardinalities of all bases are equal. This cardinality is said to be the dimension of $V$.

Isomorphism

Any two vector spaces of the same dimension are said to be isomporphous - any result obtained for one can be applied to the other.

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