"Linear space" redirects here. For a structure in incidence geometry, seeLinear space (geometry).
Vector addition and scalar multiplication: a vectorv (blue) is added to another vectorw (red, upper illustration). Below,w is stretched by a factor of 2, yielding the sumv + 2w.
Inmathematics andphysics, avector space (also called alinear space) is aset whose elements, often calledvectors, can be added together and multiplied ("scaled") by numbers calledscalars. The operations of vector addition andscalar multiplication must satisfy certain requirements, calledvectoraxioms.Real vector spaces andcomplex vector spaces are kinds of vector spaces based on different kinds of scalars:real numbers andcomplex numbers. Scalars can also be, more generally, elements of anyfield.
Vector spaces are characterized by theirdimension, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces areisomorphic). A vector space isfinite-dimensional if its dimension is anatural number. Otherwise, it isinfinite-dimensional, and its dimension is aninfinite cardinal. Finite-dimensional vector spaces occur naturally ingeometry and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example,polynomial rings arecountably infinite-dimensional vector spaces, and manyfunction spaces have thecardinality of the continuum as a dimension.
In this article, vectors are represented in boldface to distinguish them from scalars.[nb 1][1]
A vector space over afieldF is a non-emptysetV together with abinary operation and abinary function that satisfy the eightaxioms listed below. In this context, the elements ofV are commonly calledvectors, and the elements of F are calledscalars.[2]
The binary operation, calledvector addition or simplyaddition assigns to any two vectors v andw inV a third vector inV which is commonly written asv +w, and called thesum of these two vectors.
The binary function, calledscalar multiplication, assigns to any scalar a inF and any vector v inV another vector inV, which is denoted av.[nb 2]
To have a vector space, the eight followingaxioms must be satisfied for everyu,v andw inV, anda andb inF.[3]
Distributivity of scalar multiplication with respect to vector addition
a(u +v) =au +av
Distributivity of scalar multiplication with respect to field addition
(a +b)v =av +bv
When the scalar field is thereal numbers, the vector space is called areal vector space, and when the scalar field is thecomplex numbers, the vector space is called acomplex vector space.[4] These two cases are the most common ones, but vector spaces with scalars in an arbitrary fieldF are also commonly considered. Such a vector space is called anF-vector space or avector space overF.[5]
An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is anabelian group under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines aring homomorphism from the fieldF into theendomorphism ring of this group.[6]
Subtraction of two vectors can be defined as
Direct consequences of the axioms include that, for every and one has
implies or
Even more concisely, a vector space is amodule over afield.[7]
A vectorv inR2 (blue) expressed in terms of different bases: using thestandard basis ofR2:v =xe1 +ye2 (black), and using a different, non-orthogonal basis:v =f1 +f2 (red).
Given a setG of elements of aF-vector spaceV, a linear combination of elements ofG is an element ofV of the form where and The scalars are called thecoefficients of the linear combination.[8]
The elements of a subsetG of aF-vector spaceV are said to belinearly independent if no element ofG can be written as a linear combination of the other elements ofG. Equivalently, they are linearly independent if two linear combinations of elements ofG define the same element ofV if and only if they have the same coefficients. Also equivalently, they are linearly independent if a linear combination results in the zero vector if and only if all its coefficients are zero.[9]
Alinear subspace orvector subspaceW of a vector spaceV is a non-empty subset ofV that isclosed under vector addition and scalar multiplication; that is, the sum of two elements ofW and the product of an element ofW by a scalar belong toW.[10] This implies that every linear combination of elements ofW belongs toW. A linear subspace is a vector space for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied.[11] The closure property also implies thateveryintersection of linear subspaces is a linear subspace.[11]
Given a subsetG of a vector spaceV, thelinear span or simply thespan ofG is the smallest linear subspace ofV that containsG, in the sense that it is the intersection of all linear subspaces that containG. The span ofG is also the set of all linear combinations of elements ofG. IfW is the span ofG, one says thatGspans orgeneratesW, and thatG is aspanning set or agenerating set ofW.[12]
A subset of a vector space is abasis if its elements are linearly independent and span the vector space.[13] Every vector space has at least one basis, or many in general (seeBasis (linear algebra) § Proof that every vector space has a basis).[14] Moreover, all bases of a vector space have the samecardinality, which is called thedimension of the vector space (seeDimension theorem for vector spaces).[15] This is a fundamental property of vector spaces, which is detailed in the remainder of the section.
Bases are a fundamental tool for the study of vector spaces, especially when the dimension is finite. In the infinite-dimensional case, the existence of infinite bases, often calledHamel bases, depends on theaxiom of choice. It follows that, in general, no base can be explicitly described.[16] For example, thereal numbers form an infinite-dimensional vector space over therational numbers, for which no specific basis is known.
Consider a basis of a vector spaceV of dimensionn over a fieldF. The definition of a basis implies that every may be written with inF, and that this decomposition is unique. The scalars are called thecoordinates ofv on the basis. They are also said to be thecoefficients of the decomposition ofv on the basis. One also says that then-tuple of the coordinates is thecoordinate vector ofv on the basis, since the set of then-tuples of elements ofF is a vector space forcomponentwise addition and scalar multiplication, whose dimension isn.
Theone-to-one correspondence between vectors and their coordinate vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication. It is thus avector space isomorphism, which allows translating reasonings and computations on vectors into reasonings and computations on their coordinates.[17]
In 1857,Cayley introduced thematrix notation which allows for harmonization and simplification oflinear maps. Around the same time,Grassmann studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations.[24] In his work, the concepts oflinear independence anddimension, as well asscalar products are present. Grassmann's 1844 work exceeds the framework of vector spaces as well since his considering multiplication led him to what are today calledalgebras. Italian mathematicianPeano was the first to give the modern definition of vector spaces and linear maps in 1888,[25] although he called them "linear systems".[26] Peano's axiomatization allowed for vector spaces with infinite dimension, but Peano did not develop that theory further. In 1897,Salvatore Pincherle adopted Peano's axioms and made initial inroads into the theory of infinite-dimensional vector spaces.[27]
Vector addition: the sumv +w (black) of the vectorsv (blue) andw (red) is shown.
Scalar multiplication: the multiples−v and2w are shown.
The first example of a vector space consists ofarrows in a fixedplane, starting at one fixed point. This is used in physics to describeforces orvelocities.[30] Given any two such arrows,v andw, theparallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called thesum of the two arrows, and is denotedv +w. In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positivereal numbera, the arrow that has the same direction asv, but is dilated or shrunk by multiplying its length bya, is calledmultiplication ofv bya. It is denotedav. Whena is negative,av is defined as the arrow pointing in the opposite direction instead.[31]
The following shows a few examples: ifa = 2, the resulting vectoraw has the same direction asw, but is stretched to the double length ofw (the second image). Equivalently,2w is the sumw +w. Moreover,(−1)v = −v has the opposite direction and the same length asv (blue vector pointing down in the second image).
A second key example of a vector space is provided by pairs of real numbersx andy. The order of the componentsx andy is significant, so such a pair is also called anordered pair. Such a pair is written as(x,y). The sum of two such pairs and the multiplication of a pair with a number is defined as follows:[32]
The first example above reduces to this example if an arrow is represented by a pair ofCartesian coordinates of its endpoint.
The simplest example of a vector space over a fieldF is the fieldF itself with its addition viewed as vector addition and its multiplication viewed as scalar multiplication. More generally, alln-tuples (sequences of lengthn)of elementsai ofF form a vector space that is usually denotedFn and called acoordinate space.[33] The casen = 1 is the above-mentioned simplest example, in which the fieldF is also regarded as a vector space over itself. The caseF =R andn = 2 (soR2) reduces to the previous example.
The set ofcomplex numbersC, numbers that can be written in the formx +iy forreal numbersx andy wherei is theimaginary unit, form a vector space over the reals with the usual addition and multiplication:(x +iy) + (a +ib) = (x +a) +i(y +b) andc ⋅ (x +iy) = (c ⋅x) +i(c ⋅y) for real numbersx,y,a,b andc. The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. The example of complex numbers is essentially the same as (that is, it isisomorphic to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex numberx +iy as representing the ordered pair(x,y) in thecomplex plane then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example.
More generally,field extensions provide another class of examples of vector spaces, particularly in algebra andalgebraic number theory: a fieldF containing asmaller fieldE is anE-vector space, by the given multiplication and addition operations ofF.[34] For example, the complex numbers are a vector space overR, and the field extension is a vector space overQ.
Addition of functions: the sum of the sine and the exponential function is with.
Functions from any fixed setΩ to a fieldF also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functionsf andg is the function given byand similarly for multiplication. Such function spaces occur in many geometric situations, whenΩ is thereal line or aninterval, or othersubsets ofR. Many notions in topology and analysis, such ascontinuity,integrability ordifferentiability are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property.[35] Therefore, the set of such functions are vector spaces, whose study belongs tofunctional analysis.
Systems ofhomogeneous linear equations are closely tied to vector spaces.[36] For example, the solutions ofare given by triples with arbitrary and They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too.Matrices can be used to condense multiple linear equations as above into one vector equation, namely
where is the matrix containing the coefficients of the given equations, is the vector denotes thematrix product, and is the zero vector. In a similar vein, the solutions of homogeneouslinear differential equations form vector spaces. For example,
The relation of two vector spaces can be expressed bylinear map orlinear transformation. They arefunctions that reflect the vector space structure, that is, they preserve sums and scalar multiplication: for all and in all in[37]
Anisomorphism is a linear mapf :V →W such that there exists aninverse mapg :W →V, which is a map such that the two possiblecompositionsf ∘g :W →W andg ∘f :V →V areidentity maps. Equivalently,f is both one-to-one (injective) and onto (surjective).[38] If there exists an isomorphism betweenV andW, the two spaces are said to beisomorphic; they are then essentially identical as vector spaces, since all identities holding inV are, viaf, transported to similar ones inW, and vice versa viag.
Describing an arrow vectorv by its coordinatesx andy yields an isomorphism of vector spaces.
For example, the arrows in the plane and the ordered pairs of numbers vector spaces in the introduction above (see§ Examples) are isomorphic: a planar arrowv departing at theorigin of some (fixed)coordinate system can be expressed as an ordered pair by considering thex- andy-component of the arrow, as shown in the image at the right. Conversely, given a pair(x,y), the arrow going byx to the right (or to the left, ifx is negative), andy up (down, ify is negative) turns back the arrowv.[39]
Linear mapsV →W between two vector spaces form a vector spaceHomF(V,W), also denotedL(V,W), or𝓛(V,W).[40] The space of linear maps fromV toF is called thedual vector space, denotedV∗.[41] Via the injectivenatural mapV →V∗∗, any vector space can be embedded into itsbidual; the map is an isomorphism if and only if the space is finite-dimensional.[42]
Once a basis ofV is chosen, linear mapsf :V →W are completely determined by specifying the images of the basis vectors, because any element ofV is expressed uniquely as a linear combination of them.[43] IfdimV = dimW, a1-to-1 correspondence between fixed bases ofV andW gives rise to a linear map that maps any basis element ofV to the corresponding basis element ofW. It is an isomorphism, by its very definition.[44] Therefore, two vector spaces over a given field are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space over a given field iscompletely classified (up to isomorphism) by its dimension, a single number. In particular, anyn-dimensionalF-vector spaceV is isomorphic toFn. However, there is no "canonical" or preferred isomorphism; an isomorphismφ :Fn →V is equivalent to the choice of a basis ofV, by mapping the standard basis ofFn toV, viaφ.
Matrices are a useful notion to encode linear maps.[45] They are written as a rectangular array of scalars as in the image at the right. Anym-by-n matrix gives rise to a linear map fromFn toFm, by the following where denotessummation, or by using thematrix multiplication of the matrix with the coordinate vector:
Moreover, after choosing bases ofV andW,any linear mapf :V →W is uniquely represented by a matrix via this assignment.[46]
The volume of thisparallelepiped is the absolute value of the determinant of the 3-by-3 matrix formed by the vectorsr1,r2, andr3.
Thedeterminantdet (A) of asquare matrixA is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero.[47] The linear transformation ofRn corresponding to a realn-by-n matrix isorientation preserving if and only if its determinant is positive.
Endomorphisms, linear mapsf :V →V, are particularly important since in this case vectorsv can be compared with their image underf,f(v). Any nonzero vectorv satisfyingλv =f(v), whereλ is a scalar, is called aneigenvector off witheigenvalueλ.[48] Equivalently,v is an element of thekernel of the differencef −λ · Id (where Id is theidentity mapV →V). IfV is finite-dimensional, this can be rephrased using determinants:f having eigenvalueλ is equivalent toBy spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function inλ, called thecharacteristic polynomial off.[49] If the fieldF is large enough to contain a zero of this polynomial (which automatically happens forFalgebraically closed, such asF =C) any linear map has at least one eigenvector. The vector spaceV may or may not possess aneigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by theJordan canonical form of the map.[50] The set of all eigenvectors corresponding to a particular eigenvalue off forms a vector space known as theeigenspace corresponding to the eigenvalue (andf) in question.
In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones.
A line passing through theorigin (blue, thick) inR3 is a linear subspace. It is the intersection of twoplanes (green and yellow).
A nonemptysubset of a vector space that is closed under addition and scalar multiplication (and therefore contains the-vector of) is called alinear subspace of, or simply asubspace of, when the ambient space is unambiguously a vector space.[51][nb 4] Subspaces of are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called itsspan, and it is the smallest subspace of containing the set. Expressed in terms of elements, the span is the subspace consisting of all thelinear combinations of elements of.[52]
Linear subspace of dimension 1 and 2 are referred to as aline (alsovector line), and aplane respectively. IfW is ann-dimensional vector space, any subspace of dimension 1 less, i.e., of dimension is called ahyperplane.[53]
The counterpart to subspaces arequotient vector spaces.[54] Given any subspace, the quotient space ("modulo") is defined as follows: as a set, it consists ofwhere is an arbitrary vector in. The sum of two such elements and is, and scalar multiplication is given by. The key point in this definition is thatif and only if the difference of and lies in.[nb 5] This way, the quotient space "forgets" information that is contained in the subspace.
Thekernel of a linear map consists of vectors that are mapped to in.[55] The kernel and theimage are subspaces of and, respectively.[56]
An important example is the kernel of a linear map for some fixed matrix. The kernel of this map is the subspace of vectors such that, which is precisely the set of solutions to the system of homogeneous linear equations belonging to. This concept also extends to linear differential equationswhere the coefficients are functions in too.In the corresponding mapthederivatives of the function appear linearly (as opposed to, for example). Since differentiation is a linear procedure (that is, and for a constant) this assignment is linear, called alinear differential operator. In particular, the solutions to the differential equation form a vector space (overR orC).[57]
The existence of kernels and images is part of the statement that thecategory of vector spaces (over a fixed field) is anabelian category, that is, a corpus of mathematical objects and structure-preserving maps between them (acategory) that behaves much like thecategory of abelian groups.[58] Because of this, many statements such as thefirst isomorphism theorem (also calledrank–nullity theorem in matrix-related terms)and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements forgroups.
Thedirect product of vector spaces and thedirect sum of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.
Thedirect product of a family of vector spaces consists of the set of all tuples, which specify for each index in someindex set an element of.[59] Addition and scalar multiplication is performed componentwise. A variant of this construction is thedirect sum (also calledcoproduct and denoted), where only tuples with finitely many nonzero vectors are allowed. If the index set is finite, the two constructions agree, but in general they are different.
Thetensor product or simply of two vector spaces and is one of the central notions ofmultilinear algebra which deals with extending notions such as linear maps to several variables. A map from theCartesian product is calledbilinear if is linear in both variables and That is to say, for fixed the map is linear in the sense above and likewise for fixed
The tensor product is a particular vector space that is auniversal recipient of bilinear maps as follows. It is defined as the vector space consisting of finite (formal) sums of symbols calledtensorssubject to the rules[60]These rules ensure that the map from the to that maps atuple to is bilinear. The universality states that givenany vector space andany bilinear map there exists a unique map shown in the diagram with a dotted arrow, whosecomposition with equals[61] This is called theuniversal property of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.
From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space over a given field is characterized, up to isomorphism, by its dimension. However, vector spacesper se do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functionsconverges to another function. Likewise, linear algebra is not adapted to deal withinfinite series, since the addition operation allows only finitely many terms to be added.Therefore, the needs offunctional analysis require considering additional structures.[62]
A vector space may be given apartial order under which some vectors can be compared.[63] For example,-dimensional real space can be ordered by comparing its vectors componentwise.Ordered vector spaces, for exampleRiesz spaces, are fundamental toLebesgue integration, which relies on the ability to express a function as a difference of two positive functionswhere denotes the positive part of and the negative part.[64]
"Measuring" vectors is done by specifying anorm, a datum which measures lengths of vectors, or by aninner product, which measures angles between vectors. Norms and inner products are denoted and respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm Vector spaces endowed with such data are known asnormed vector spaces andinner product spaces, respectively.[65]
Coordinate space can be equipped with the standarddot product:In this reflects the common notion of the angle between two vectors and by thelaw of cosines:Because of this, two vectors satisfying are calledorthogonal. An important variant of the standard dot product is used inMinkowski space: endowed with the Lorentz product[66]In contrast to the standard dot product, it is notpositive definite: also takes negative values, for example, for Singling out the fourth coordinate—corresponding to time, as opposed to three space-dimensions—makes it useful for the mathematical treatment ofspecial relativity. Note that in other conventions time is often written as the first, or "zeroeth" component so that the Lorentz product is written
Convergence questions are treated by considering vector spaces carrying a compatibletopology, a structure that allows one to talk about elements beingclose to each other.[67] Compatible here means that addition and scalar multiplication have to becontinuous maps. Roughly, if and in, and in vary by a bounded amount, then so do and[nb 6] To make sense of specifying the amount a scalar changes, the field also has to carry a topology in this context; a common choice is the reals or the complex numbers.
In suchtopological vector spaces one can considerseries of vectors. Theinfinite sumdenotes thelimit of the corresponding finite partial sums of the sequence of elements of For example, the could be (real or complex) functions belonging to somefunction space in which case the series is afunction series. Themode of convergence of the series depends on the topology imposed on the function space. In such cases,pointwise convergence anduniform convergence are two prominent examples.[68]
Unit "spheres" in consist of plane vectors of norm 1. Depicted are the unit spheres in different-norms, for and The bigger diamond depicts points of 1-norm equal to 2.
A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where anyCauchy sequence has a limit; such a vector space is calledcomplete. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval equipped with thetopology of uniform convergence is not complete because any continuous function on can be uniformly approximated by a sequence of polynomials, by theWeierstrass approximation theorem.[69] In contrast, the space ofall continuous functions on with the same topology is complete.[70] A norm gives rise to a topology by defining that a sequence of vectors converges to if and only ifBanach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece offunctional analysis—focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence.[71] The image at the right shows the equivalence of the-norm and-norm on as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data.
From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also calledfunctionals) maps between topological vector spaces are required to be continuous.[72] In particular, the(topological) dual space consists of continuous functionals (or to). The fundamentalHahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.[73]
A first example isthe vector space consisting of infinite vectors with real entrieswhose-norm given by
The topologies on the infinite-dimensional space are inequivalent for different For example, the sequence of vectors in which the first components are and the following ones are converges to thezero vector for but does not for but
More generally than sequences of real numbers, functions are endowed with a norm that replaces the above sum by theLebesgue integral
These spaces are complete.[75] (If one uses theRiemann integral instead, the space isnot complete, which may be seen as a justification for Lebesgue's integration theory.[nb 8]) Concretely this means that for any sequence of Lebesgue-integrable functions with satisfying the conditionthere exists a function belonging to the vector space such that
The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).
Complete inner product spaces are known asHilbert spaces, in honor ofDavid Hilbert.[77] The Hilbert space with inner product given bywhere denotes thecomplex conjugate of[78][nb 9] is a key case.
By definition, in a Hilbert space, any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions with desirable properties that approximate a given limit function is equally crucial. Early analysis, in the guise of theTaylor approximation, established an approximation ofdifferentiable functions by polynomials.[79] By theStone–Weierstrass theorem, every continuous function on can be approximated as closely as desired by a polynomial.[80] A similar approximation technique bytrigonometric functions is commonly calledFourier expansion, and is much applied in engineering. More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space in the sense that theclosure of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called abasis of its cardinality is known as theHilbert space dimension.[nb 10] Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with theGram–Schmidt process, it enables one to construct abasis of orthogonal vectors.[81] Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensionalEuclidean space.
The solutions to variousdifferential equations can be interpreted in terms of Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations, and frequently solutions with particular physical properties are used as basis functions, often orthogonal.[82] As an example from physics, the time-dependentSchrödinger equation inquantum mechanics describes the change of physical properties in time by means of apartial differential equation, whose solutions are calledwavefunctions.[83] Definite values for physical properties such as energy, or momentum, correspond toeigenvalues of a certain (linear)differential operator and the associated wavefunctions are calledeigenstates. Thespectral theorem decomposes a linearcompact operator acting on functions in terms of these eigenfunctions and their eigenvalues.[84]
Ahyperbola, given by the equation Thecoordinate ring of functions on this hyperbola is given by an infinite-dimensional vector space over
General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additionalbilinear operator defining the multiplication of two vectors is analgebra over a field (orF-algebra if the fieldF is specified).[85]
Another crucial example areLie algebras, which are neither commutative nor associative, but the failure to be so is limited by the constraints ( denotes the product of and):
Examples include the vector space of-by- matrices, with thecommutator of two matrices, and endowed with thecross product.
Thetensor algebra is a formal way of adding products to any vector space to obtain an algebra.[88] As a vector space, it is spanned by symbols, called simpletensorswhere thedegree varies.The multiplication is given by concatenating such symbols, imposing thedistributive law under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced in the above section ontensor products. In general, there are no relations between and Forcing two such elements to be equal leads to thesymmetric algebra, whereas forcing yields theexterior algebra.[89]
Avector bundle is a family of vector spaces parametrized continuously by atopological spaceX.[90] More precisely, a vector bundle overX is a topological spaceE equipped with a continuous mapsuch that for everyx inX, thefiber π−1(x) is a vector space. The case dimV = 1 is called aline bundle. For any vector spaceV, the projectionX ×V →X makes the productX ×V into a"trivial" vector bundle. Vector bundles overX are required to belocally a product ofX and some (fixed) vector spaceV: for everyx inX, there is aneighborhoodU ofx such that the restriction of π to π−1(U) is isomorphic[nb 11] to the trivial bundleU ×V →U. Despite their locally trivial character, vector bundles may (depending on the shape of the underlying spaceX) be "twisted" in the large (that is, the bundle need not be (globally isomorphic to) the trivial bundleX ×V). For example, theMöbius strip can be seen as a line bundle over the circleS1 (byidentifying open intervals with the real line). It is, however, different from thecylinderS1 ×R, because the latter isorientable whereas the former is not.[91]
Properties of certain vector bundles provide information about the underlying topological space. For example, thetangent bundle consists of the collection oftangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circleS1 is globally isomorphic toS1 ×R, since there is a global nonzerovector field onS1.[nb 12] In contrast, by thehairy ball theorem, there is no (tangent) vector field on the2-sphereS2 which is everywhere nonzero.[92]K-theory studies the isomorphism classes of all vector bundles over some topological space.[93] In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional realdivision algebras:R,C, thequaternionsH and theoctonionsO.
Modules are torings what vector spaces are to fields: the same axioms, applied to a ringR instead of a fieldF, yield modules.[94] The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not havemultiplicative inverses. For example, modules need not have bases, as theZ-module (that is,abelian group)Z/2Z shows; those modules that do (including all vector spaces) are known asfree modules. Nevertheless, a vector space can be compactly defined as amodule over aring which is afield, with the elements being called vectors. Some authors use the termvector space to mean modules over adivision ring.[95] The algebro-geometric interpretation of commutative rings via theirspectrum allows the development of concepts such aslocally free modules, the algebraic counterpart to vector bundles.
Anaffine plane (light blue) inR3. It is a two-dimensional subspace shifted by a vectorx (red).
Roughly,affine spaces are vector spaces whose origins are not specified.[96] More precisely, an affine space is a set with afree transitive vector spaceaction. In particular, a vector space is an affine space over itself, by the mapIfW is a vector space, then an affine subspace is a subset ofW obtained by translating a linear subspaceV by a fixed vectorx ∈W; this space is denoted byx +V (it is acoset ofV inW) and consists of all vectors of the formx +v forv ∈V. An important example is the space of solutions of a system of inhomogeneous linear equationsgeneralizing the homogeneous case discussed in theabove section on linear equations, which can be found by setting in this equation.[97] The space of solutions is the affine subspacex +V wherex is a particular solution of the equation, andV is the space of solutions of the homogeneous equation (thenullspace ofA).
The set of one-dimensional subspaces of a fixed finite-dimensional vector spaceV is known asprojective space; it may be used to formalize the idea ofparallel lines intersecting at infinity.[98]Grassmannians andflag manifolds generalize this by parametrizing linear subspaces of fixed dimensionk andflags of subspaces, respectively.
^It is also common, especially in physics, to denote vectors with an arrow on top: It is also common, especially in higher mathematics, to not use any typographical method for distinguishing vectors from other mathematical objects.
^Scalar multiplication is not to be confused with thescalar product, which is an additional operation on some specific vector spaces, calledinner product spaces. Scalar multiplication is the multiplication of a vectorby a scalar that produces a vector, while the scalar product is a multiplication of two vectors that produces a scalar.
^This axiom is not anassociative property, since it refers to two different operations, scalar multiplication and field multiplication. So, it is independent from the associativity of field multiplication, which is assumed by field axioms.
^This is typically the case when a vector space is also considered as anaffine space. In this case, a linear subspace contains thezero vector, while an affine subspace does not necessarily contain it.
^"Many functions in of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration.",Dudley (1989), §5.3, p. 125.
^A basis of a Hilbert space is not the same thing as a basis of a linear algebra. For distinction, a linear algebra basis for a Hilbert space is called aHamel basis.
^That is, there is ahomeomorphism from π−1(U) toV ×U which restricts to linear isomorphisms between fibers.
^A line bundle, such as the tangent bundle ofS1 is trivial if and only if there is asection that vanishes nowhere, seeHusemoller (1994), Corollary 8.3. The sections of the tangent bundle are justvector fields.
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