In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, science, and engineering. Linear algebra is the branch of Mathematics concerned with

The most familiar vector spaces are two- and three-dimensional Euclidean spaces. Vectors in these spaces can be represented by ordered pairs or triples of real numbers, and are isomorphic to geometric vectors—quantities with a magnitude and a direction, usually depicted as arrows. In Mathematics, the real numbers may be described informally in several different ways In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective These vectors may be added together using the parallelogram rule (vector addition) or multiplied by real numbers (scalar multiplication). In Mathematics, the simplest form of the parallelogram law belongs to elementary Geometry. In Mathematics, scalar multiplication is one of the basic operations defining a Vector space in Linear algebra (or more generally a module in 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. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

A vector space is a set of objects (called vectors) that can be scaled and added.

Formal definition

Let F be a field (such as the real numbers or complex numbers), whose elements will be called scalars. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication A vector space over the field F is a set V together with two binary operations,

• vector addition: V × V → V denoted v + w, where v, w ∈ V, and
• scalar multiplication: F × V → V denoted av, where a ∈ F and v ∈ V,

satisfying the axioms below. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Four of the axioms require vectors under addition to form an abelian group, and two are distributive laws. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law

1. Vector addition is associative:

   For all u, v, w ∈ V, 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, w ∈ V, we have v + w = w + v. In Mathematics, commutativity is the ability to change the order of something without changing the end result

3. Vector addition has an identity element:

   There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V. 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 In Linear algebra, the null vector or zero vector is the vector (0 0 &hellip 0 in Euclidean space, all of whose components are zero

4. Vector addition has inverse elements:

   For all v ∈ V, there exists an element w ∈ V, called the additive inverse of v, such that v + w = 0. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In mathematics the additive inverse, or opposite, of a Number n is the number that when added to n, yields zero.

5. Distributivity holds for scalar multiplication over vector addition:

   For all a ∈ F and v, w ∈ V, we have a (v + w) = a v + a w.

6. Distributivity holds for scalar multiplication over field addition:

   For all a, b ∈ F and v ∈ V, we have (a + b) v = a v + b v.

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

   For all a, b ∈ F and v ∈ V, we have a (b v) = (ab) v.

8. Scalar multiplication has an identity element:

   For all v ∈ V, we have 1 v = v, where 1 denotes the multiplicative identity in F. 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 Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity

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

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:

1. V is closed under vector addition:

   If u, v ∈ V, then u + v ∈ V. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set

2. V is closed under scalar multiplication:

   If a ∈ F, v ∈ V, then a v ∈ V. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set

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. The validity of closure axioms is key to determining whether a subset of a vector space is a subspace. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics.

Note that expressions of the form “v a”, where v ∈ V and a ∈ F, are, strictly speaking, not defined. Because of the commutativity of the underlying field, however, “a v” and “v a” are often treated synonymously. Additionally, if v ∈ V, w ∈ V, and a ∈ F where vector space V is additionally an algebra over the field F then a v w = v a w, which makes it convenient to consider “a v” and “v a” to represent the same vector. In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with

Elementary properties

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

• The zero vector 0 ∈ V is unique:

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

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

  For all a ∈ F, we have a 0 = 0.

• Scalar multiplication by zero yields the zero vector:

  For all v ∈ V, 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 v ∈ V, such that v + w₁ = 0 and v + w₂ = 0, then w₁ = w₂. We call the inverse −v and define w − v ≡ w + (−v).

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

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

• Negation commutes freely:

  For all a ∈ F and v ∈ V, we have (−a) v = a (−v) = − (a v).

Examples

Main article: Examples of vector spaces

Subspaces and bases

Main articles: Linear subspace, Basis

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. This page lists some examples of vector spaces. See Vector space for the definitions of terms used on this page The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. 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. In the mathematical subfield of Linear algebra, the linear span, also called the linear hull, of a set of vectors in a Vector In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors A linearly independent set whose span is V is called a basis for V. Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

Using Zorn’s Lemma (which is equivalent to the axiom of choice), it can be proven that every vector space has a basis. Zorn's lemma, also known as the Kuratowski-Zorn lemma, is a proposition of Set theory that states Every Partially ordered set in which In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. 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. In Mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given Abstract algebra. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" 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. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In Mathematics, the dimension of a Vector space V is the cardinality (i For instance, the real finite-dimensional vector spaces are just R⁰, R¹, R², R³, …. The dimension of the real vector space R³ is three.

It was F. Hausdorff who first proved that every vector space has a basis. Andreas Blass^[1] showed this theorem leads to the axiom of choice. Andreas Raphael Blass (born on October 27 1947 in Nuremberg) is a mathematician currently a professor at the University of Michigan. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

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. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Vector spaces are sometimes introduced from this coordinatised viewpoint.

One often considers vector spaces which also carry a compatible topology. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. 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. In the Mathematical field of Topology, a uniform space is a set with 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.

Only in such topological vector spaces can one consider infinite sums of vectors, i. In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. e. series, through the notion of convergence. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with 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 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. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions

Linear maps

Main article: Linear map

Given two vector spaces V and W over the same field F, one can define linear maps or “linear transformations” from V to W. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that These are functions f:V → W that are compatible with the relevant structure — i. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function e. , they preserve sums and scalar products. The set of all linear maps from V to W, denoted Hom_F (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 matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

An isomorphism is a linear map such that there exists an inverse map such that and are identity maps. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics and related technical fields the term map or mapping is often a Synonym for function. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that A linear map that is both one-to-one (injective) and onto (surjective) is necessarily an isomorphism. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every 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.

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 it abstracts from sets In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist

Generalizations

From an abstract point of view, vector spaces are modules over a field, F. 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 The common practice of identifying a v and v a in a vector space makes the vector space an F-F bimodule. 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 Modules in general need not have bases; those that do (including all vector spaces) are known as free modules. In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction

A family of vector spaces, parametrised continuously by some underlying topological space, is a vector bundle. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space

An affine space is a set with a transitive vector space action. In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. Note that a vector space is an affine space over itself, by the structure map

Additional structures

It is common to study vector spaces with certain additional structures. In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. This is often necessary for recovering ordinary notions from geometry.

• A real or complex vector space with a well-defined concept of length, i. e. , a norm, is called a normed vector space. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length 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
• A normed vector space with the additional well-defined concept of angle is called an inner product space. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.
• A vector space with a topology compatible with the operations — such that addition and scalar multiplication are continuous maps — is called a topological vector space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. The topological structure is relevant when the underlying vector space is infinite dimensional.

• A vector space with an additional bilinear operator defining the multiplication of two vectors is an algebra over a field. In Mathematics, a bilinear map is a function of two arguments that is linear in each In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with
• An ordered vector space. In Mathematics an ordered vector space or partially ordered vector space is a Vector space equipped with a Partial order which is compatible

Notes

1. ^ Blass, Andreas (1984), “Existence of bases implies the axiom of choice”, Axiomatic set theory, Proc. Andreas Raphael Blass (born on October 27 1947 in Nuremberg) is a mathematician currently a professor at the University of Michigan. AMS-IMS-SIAM Jt. Summer Res. Conf. , Boulder/Colo. 1983, Contemp. Math. 31, 31-33. , pp. 31–34 

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

• Vector (spatial), for vectors in physics
• Vector field
• Vector spaces without fields