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Encyclopedia > Vector space

In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are defined and satisfy certain natural axioms which are listed below. Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, science, and engineering. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ...


The most familiar vector spaces are two- and three-dimensional Euclidean spaces. Vectors in these spaces are ordered pairs or triples of real numbers, and are often represented as geometric vectors which are quantities with a magnitude and a direction, usually depicted as arrows. These vectors may be added together using the parallelogram rule (vector addition) or multiplied by real numbers (scalar multiplication). The behavior of geometric vectors under these operations provides a good intuitive model for the behavior of vectors in more abstract vector spaces, which need not have a geometric interpretation. For example, the set of (real) polynomials forms a vector space. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the real numbers may be described informally in several different ways. ... This article is about vectors that have a particular relation to the spatial coordinates. ... In mathematics, the simplest form of the parallelogram law belongs to elementary geometry. ...

Contents

Formal definition

Let F be a field (such as the real numbers or complex numbers), whose elements will be called scalars. A vector space over the field F is a set V together with two binary operations, In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

  • vector addition: V × VV denoted v + w, where v, wV, and
  • scalar multiplication: F × VV denoted a v, where aF and vV,

satisfying the axioms below. Four require vector addition to be an Abelian group, and two are distributive laws. For the algebra software named Axiom, see Axiom computer algebra system. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

  1. Vector addition is associative:

    For all u, v, wV, we have u + (v + w) = (u + v) + w. In mathematics, associativity is a property that a binary operation can have. ...

  2. Vector addition is commutative:

    For all v, wV, we have v + w = w + v. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

  3. Vector addition has an identity element:

    There exists an element 0V, called the zero vector, such that v + 0 = v for all vV. In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition. ...

  4. Vector addition has an inverse element:

    For all v ∈ V, there exists an element wV, called the additive inverse of v, such that v + w = 0. In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ...

  5. Distributivity holds for scalar multiplication over vector addition:

    For all aF and v, wV, we have a (v + w) = a v + a w.

  6. Distributivity holds for scalar multiplication over field addition:

    For all a, bF and vV, we have (a + b) v = a v + b v.

  7. Scalar multiplication is compatible with multiplication in the field of scalars:

    For all a, bF and vV, we have a (b v) = (ab) v.

  8. Scalar multiplication has an identity element:

    For all vV, we have 1 v = v, where 1 denotes the multiplicative identity in F. One redirects here. ...

Formally, these are the axioms for a module, so a vector space may be concisely described as a module over a field. In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...


Note that the seventh axiom above, stating a (b v) = (ab) v, is not asserting the associativity of an operation, since there are two operations in question, scalar multiplication: b v; and field multiplication: ab. In mathematics, associativity is a property that a binary operation can have. ...


Some sources choose to also include two axioms of closure: In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...

  1. V is closed under vector addition:

    If u, vV, then u + vV. In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...

  2. V is closed under scalar multiplication:

    If aF, vV, then a vV. In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...

However, the modern formal understanding of the operations as maps with codomain V implies these statements by definition, and thus obviates the need to list them as independent axioms.


Note that expressions of the form “v a”, where vV and aF, are, strictly speaking, not defined. Because of the commutativity of the underlying field, however, “a v” and “v a” may be treated synonymously, and this is often done in practice.


Elementary properties

There are a number of properties that follow easily from the vector space axioms.

  • The zero vector 0V is unique:

    If 01 and 02 are zero vectors in V, such that 01 + v = v and 02 + v = v for all vV, then 01 = 02 = 0.

  • Scalar multiplication with the zero vector yields the zero vector:

    For all aF, we have a 0 = 0.

  • Scalar multiplication by zero yields the zero vector:

    For all vV, we have 0 v = 0, where 0 denotes the additive identity in F.

  • No other scalar multiplication yields the zero vector:

    We have a v = 0 if and only if a = 0 or v = 0.

  • The additive inverse −v of a vector v is unique:

    If w1 and w2 are additive inverses of vV, such that v + w1 = 0 and v + w2 = 0, then w1 = w2. We call the inverse −v and define w − vw + (−v).

  • Scalar multiplication by negative unity yields the additive inverse of the vector:

    For all vV, we have (−1) v = −v, where 1 denotes the multiplicative identity in F.

  • Negation commutes freely:

    For all aF and vV, we have (−a) v = a (−v) = − (a v).

Examples

See Examples of vector spaces for a list of standard examples. This page lists some examples of vector spaces. ...


Subspaces and bases

Main articles: Linear subspace, Basis The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...


Given a vector space V, a nonempty subset W of V that is closed under addition and scalar multiplication is called a subspace of V. Subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called its span; if no vector can be removed without changing the span, the set is said to be linearly independent. A linearly independent set whose span is V is called a basis for V. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...


Using Zorn’s Lemma (which is equivalent to the axiom of choice), it can be proven that every vector space has a basis. It follows from the ultrafilter lemma, which is weaker than the axiom of choice, that all bases of a given vector space have the same cardinality. Thus vector spaces over a given field are fixed up to isomorphism by a single cardinal number (called the dimension of the vector space) representing the size of the basis. For instance, the real finite-dimensional vector spaces are just R0, R1, R2, R3, …. The dimension of the real vector space R3 is three. Zorns lemma, also known as the Kuratowski-Zorn lemma, is a proposition of set theory that states: Every non-empty partially ordered set in which every chain (i. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In mathematics, the Ultrafilter Lemma states that every filter is a subset of some ultrafilter, i. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... In mathematics, the dimension of a vector space V is the cardinality (i. ...


A basis makes it possible to express every vector of the space as a unique tuple of the field elements, although caution must be exercised when a vector space does not have a finite basis. Vector spaces are sometimes introduced from this coordinatised viewpoint. In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...


One often considers vector spaces which also carry a compatible topology. Compatible here means that addition and scalar multiplication should be continuous operations. This requirement actually ensures that the topology gives rise to a uniform structure. When the dimension is infinite, there is generally more than one inequivalent topology, which makes the study of topological vector spaces richer than that of general vector spaces. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ...


Only in such a topological vector spaces can one consider infinite sums of vectors, i.e. series, through the notion of convergence. This is of importance in both pure- and applied mathematics, for instance in quantum mechanics, where physical systems are defined as Hilbert spaces, or where Fourier expansions are used. In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ... Fig. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, a Fourier series, named in honor of Joseph Fourier (1768-1830), is a representation of a periodic function (often taken to have period 2π — in a sense, the simplest case) as a sum of periodic functions of the form which are harmonics of ei x. ...


Linear transformations

Main article: Linear transformation In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...


Given two vector spaces V and W over the same field F, one can define linear transformations or “linear maps” from V to W. These are functions f:VW that are compatible with the relevant structure — i.e., they preserve sums and scalar products. The set of all linear maps from V to W, denoted HomF (V, W), is also a vector space over F. When bases for both V and W are given, linear maps can be expressed in terms of components as matrices. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... Partial plot of a function f. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...


An isomorphism is a linear map f:Vto W such that there exists an inverse map g:Wto V such that f circ g:W to W and g circ f:V to V are identity maps. A linear map that is both one-to-one (injective) and onto (surjective) is necessarily an isomorphism. If there exists an isomorphism between V and W, the two spaces are said to be isomorphic; they are then essentially identical as vector spaces. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...


The vector spaces over a fixed field F together with the linear maps are a category, indeed an abelian category. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ...


Generalizations

From an abstract point of view, vector spaces are modules over a field, F. The common practice of identifying a v and v a in a vector space makes the vector space an F-F bimodule. Modules in general need not have bases; those that do (including all vector spaces) are known as free modules. In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. ... In mathematics, a free module is a module having a free basis. ...


A family of vector spaces, parametrised continuously by some underlying topological space, is a vector bundle. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...


An affine space is a set with a transitive vector space action. Note that a vector space is an affine space over itself, by the structure map In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ...

Theta : V^2 to V : (a, b) mapsto Theta(a, b) =: a - b,

Additional structures

It is common to study vector spaces with certain additional structures. This is often necessary for recovering ordinary notions from geometry.

In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...

References

  • Howard Anton and Chris Rorres. Elementary Linear Algebra, Wiley, 9th edition, ISBN 0-471-66959-8.
  • Kenneth Hoffmann and Ray Kunze. Linear Algebra, Prentice Hall, ISBN 0-13-536797-2.
  • Seymour Lipschutz and Marc Lipson. Schaum's Outline of Linear Algebra, McGraw-Hill, 3rd edition, ISBN 0-07-136200-2.
  • Gregory H. Moore. The axiomatization of linear algebra: 1875-1940, Historia Mathematica 22 (1995), no. 3, 262-303.
  • Gilbert Strang. "Introduction to Linear Algebra, Third Edition", Wellesley-Cambridge Press, ISBN 0-9614088-9-8

See also

Wikibooks
Wikibooks Algebra has more about this subject:
Vector spaces

  Results from FactBites:
 
Vector space - Wikipedia, the free encyclopedia (1420 words)
In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors.
Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, the sciences, and engineering.
Vectors in these spaces are ordered pairs or triples of real numbers, and are often represented as geometric vectors (quantities with a magnitude and a direction, usually depicted as arrows).
Topological vector space - Wikipedia, the free encyclopedia (1065 words)
As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a vector space.
A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by −1).
Vector addition and scalar multiplication are not only continuous but even homeomorphic which means we can construct a base for the topology and thus reconstruct the whole topology of the space from any local base around the origin.
  More results at FactBites »

 
 

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