Difference between revisions of "Vector space"
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* The cardinality of an independent subset can never exceed that of a generating subset. | * 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>. | 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>. | ||
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+ | == Isomorphism == | ||
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+ | Any two vector spaces of the same dimension are said to be isomporphous - any result obtained for one can be applied to the other. |
Revision as of 14:14, 4 November 2006
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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 over a field consists of a set (of vectors) and two operations, vector addition and scalar multiplication, which obey the following rules:
Contents
[hide]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.
- Scalar multiplication is associative, so if and then .
- Scalar multiplication is distributive over both vector and scalar addition, so if 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 .
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.