Difference between revisions of "Vector space"
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== Independent Subsets == | == Independent Subsets == | ||
| − | Let <math>V</math> be | + | Let <math>V</math> be any vector space. Let <math>I</math> be a subset of <math>V</math> such that no linear combination of elements of <math>I</math> with coefficients not all zero gives the null vector. Then <math>I</math> is said to be a linearly independent subset of <math>V</math>. An independent subset is said to be maximal if on adding any other element it ceases to be independent. |
== Linear Manifolds == | == Linear Manifolds == | ||
| Line 28: | Line 28: | ||
== Generating Subset == | == Generating Subset == | ||
| − | If <math>X</math> is a subset of a vector space <math>V</math>, such that <math>M(X) = V</math>, <math>X</math> is said to be a generating subset of <math>V</math>. | + | If <math>X</math> is a subset of a vector space <math>V</math>, such that <math>M(X) = V</math>, <math>X</math> is said to be a generating subset of <math>V</math>. 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 <math>V</math> 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 <math>V</math>. | ||
Revision as of 14:10, 4 November 2006
This article is a stub. Help us out by expanding it.
A vector space over a field (frequently the real numbers) is an object which arises in linear algebra and abstract algebra. A vector space 
over a field 
consists of a set (of vectors) and two operations, vector addition and scalar multiplication, which obey the following rules:
Contents
Axioms of a vector space
- Under vector addition, the set of vectors forms an abelian group. Thus, addition is associative and commutative and there is an additive identity (usually denoted
) and additive inverses.
) and additive - Scalar multiplication is associative, so if
and 
then 
.
and 
then 
.- Scalar multiplication is distributive over both vector and scalar addition, so if
and 
then 
.
and 
then 
.- if
, 
, 
Subspaces
If 
, and 
is a vector space itself, then it is called a subspace of
.
Independent Subsets
Let be any vector space. Let 
be a subset of such that no linear combination of elements of with coefficients not all zero gives the null vector. Then is said to be a linearly independent subset of . An independent subset is said to be maximal if on adding any other element it ceases to be independent.
Linear Manifolds
Let 
be a subset of some vector space . Then it can be proved that the set of all linear combinations of the elements of forms a vector space. This space is said to have been generated by , and is called the linear manifold 
of .
Generating Subset
If is a subset of a vector space , such that 
, is said to be a generating subset of . 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 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 .
