Difference between revisions of "Vector space"
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A '''vector space''' over a [[field]] (frequently the [[real number]]s) is an object which arises in [[linear algebra]] and [[abstract algebra]]. A vector space <math>V</math> over a field <math>F</math> consists of a [[set]] (of [[vector]]s) and two operations, vector addition and [[scalar]] multiplication, which obey the following rules: | A '''vector space''' over a [[field]] (frequently the [[real number]]s) is an object which arises in [[linear algebra]] and [[abstract algebra]]. A vector space <math>V</math> over a field <math>F</math> consists of a [[set]] (of [[vector]]s) and two operations, vector addition and [[scalar]] multiplication, which obey the following rules: | ||
− | + | ==Axioms of 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 <math>\mathbf 0</math>) and additive [[inverse with respect to an operation | inverses]]. | * 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 <math>\mathbf 0</math>) and additive [[inverse with respect to an operation | inverses]]. | ||
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* Scalar multiplication is associative, so if <math>r, s \in F</math> and <math>{\mathbf v} \in V</math> then <math>(rs){\mathbf v} = r(s{\mathbf v})</math>. | * Scalar multiplication is associative, so if <math>r, s \in F</math> and <math>{\mathbf v} \in V</math> then <math>(rs){\mathbf v} = r(s{\mathbf v})</math>. | ||
− | * Scalar multiplication | + | * Scalar multiplication [[distributive property | distributes]] over vector addition, so if <math>r \in F</math> and <math>\mathbf{v, w} \in V</math> then <math>r(\mathbf{v + w}) = r{\mathbf v} + r{\mathbf w}</math>. |
− | * | + | * Scalar multiplication by the multiplicative identity of <math>F</math> is the identity transformation, so <math>\forall {\mathbf x} \in V</math>, <math>1\cdot{\mathbf x}={\mathbf x}</math> |
== Subspaces == | == Subspaces == | ||
− | If <math>S \subseteq V</math> | + | If <math>S \subseteq V</math> and <math>S</math> is a vector space itself (over the same field), then it is called a ''subspace'' of <math>V</math>. |
− | <math>V</math>. | ||
== Independent Subsets == | == Independent Subsets == | ||
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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. | 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. | ||
− | == | + | == Span == |
− | Let <math>X</math> be a subset of some vector space <math>V</math>. Then | + | Let <math>X</math> be a subset of some vector space <math>V</math>. Then the set of all linear combinations of the elements of <math>X</math> forms a subspace of <math>V</math>. This space is said to have been generated by <math>X</math>, and is called the ''span'' of <math>X</math>. |
== Generating Subset == | == Generating Subset == | ||
− | If <math>X</math> is a subset of a vector space <math>V</math> | + | If <math>X</math> is a subset of a vector space <math>V</math> such that <math>\textrm{span}(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 == | == Basis and dimension == | ||
The following statements can be proved using the above definitions: | The following statements can be proved using the above definitions: | ||
− | * All minimal generating subsets have the same cardinality. | + | * All minimal generating subsets have the same [[cardinality]]. |
* All maximal independent 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. | * 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. | + | |
+ | 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. The cardinalities of all bases are equal. This cardinality is said to be the ''dimension'' of <math>V</math>. | ||
== Isomorphism == | == Isomorphism == | ||
− | Any two vector spaces of the same dimension are | + | Any two vector spaces of the same dimension over the same field are isomorphic -- there exists a [[bijection]] between the vector spaces which commutes with scalar multiplication and vector addition. Two isomorphic vector spaces are in some sense "the same," and any fact about one should also be true of the other. |
+ | |||
+ | [[Category:Definition]] | ||
+ | [[Category:Geometry]] | ||
+ | [[Category:Abstract algebra]] | ||
+ | [[Category:Linear algebra]] |
Latest revision as of 00:47, 21 March 2009
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 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 distributes over vector addition, so if
and
then
.
- Scalar multiplication by the multiplicative identity of
is the identity transformation, so
,
Subspaces
If and
is a vector space itself (over the same field), 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.
Span
Let be a subset of some vector space
. Then the set of all linear combinations of the elements of
forms a subspace of
. This space is said to have been generated by
, and is called the span 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. 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 over the same field are isomorphic -- there exists a bijection between the vector spaces which commutes with scalar multiplication and vector addition. Two isomorphic vector spaces are in some sense "the same," and any fact about one should also be true of the other.