Difference between revisions of "Vector space"
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Let <math>X</math> be a subset of some vector space <math>V</math>. Then it can be proved that the set of all linear combinations of the elements of <math>X</math> forms a vector space. This space is said to have been generated by <math>X</math>, and is called the linear manifold <math>M(X)</math> of <math>X</math>. | Let <math>X</math> be a subset of some vector space <math>V</math>. Then it can be proved that the set of all linear combinations of the elements of <math>X</math> forms a vector space. This space is said to have been generated by <math>X</math>, and is called the linear manifold <math>M(X)</math> of <math>X</math>. | ||
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+ | == Generating Subset == | ||
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+ | 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>. |
Revision as of 13:59, 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 a vector space over the complex field. 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 .
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 .