Vector space
In mathematics, physics, and engineering, a vector space (also called a linear space) is a set whose elements, often called "vectors", may be added together and multiplied ("scaled") by numbers called "scalars". Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called "vector axioms". The terms real vector space and complex vector space are often used to specify the nature of the scalars. Vector spaces generalize Euclidean vectors, which allow modeling physical quantities, such as forces and velocity , that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equations. Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces with the same dimension, the properties that depend only from the vector-space structure are exactly the same (technically the vector spaces are isomorphic). A vector space  is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example, polynomial rings are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension. Many vector spaces that are considered in mathematics are also endowed with other structures. This is the case of algebras, which include field extensions, polynomial rings, associative algebras and Lie algebras. This is also  the case of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces. Definition and basic properties. In this article, vectors are represented in boldface to distinguish them from scalars. A vector space over a field is a set  together with two binary operations that satisfy the eight axioms listed below. In this context, the elements of are commonly called "vectors", and the elements of  are called "scalars". For having a vector space these two operations must satisfy the eight following axioms that must be satisfied for every , and in , and and in  . When the scalar field is the real numbers the vector space is called a "real vector space". When the scalar field is the complex numbers, the vector space is called a "complex vector space". These two cases are the most common ones, but vector spaces with scalars in an arbitrary field are also commonly considered. Such a vector space is called an -"vector space" or a "vector space over ". An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms say that a vector space is an abelian group  under addition, and the four remainder axioms say that the scalar multiplication defines a ring homomorphism from the field into the endomorphism ring of this group. Subtraction of two vectors can be defined as Direct consequences of the axioms include that, for every formula_2 and formula_3 one has Related concepts and properties. "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 called Hamel bases, depend on the axiom of choice. It follows that, in general, no base can be explicitly described . For example, the real numbers form an infinite-dimensional vector space over the rational numbers, for which no specific basis is known. Let consider a basis formula_14 of a vector space of dimension over a field . The definition of a basis implies that every formula_15 may be written with formula_17 in , and that this decomposition is unique. The scalars formula_18 are called the "coordinates" of on the basis. They are also said to be the "coefficients" of the decomposition of on the basis. One says also that the -tuple of the coordinates is the coordinate vector of on the basis , since the set formula_19 of the -tuples of elements of is a vector space for componentwise addition and scalar multiplication, whose dimension is . The one-to-one correspondence between vectors and their coordinates vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication. It is thus a vector space isomorphism, which allows translating reasonings and computations on vectors into reasonings and computations on their coordinates. In turn, if these coordinates are arranged as matrices, these reasonings and computations on coordinates can be expressed concisely as reasonings and computations on matrices. Moreover, a linear equation relating matrices can  be expanded into a system of linear equations, and, conversely, every such system can be compacted into a linear equation on matrixes. So, in summary, finite-dimensional linear algebra may be expressed in three equivalent languages: History. Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three-dimensional space. Around 1636, French mathematicians René Descartes and Pierre de Fermat founded analytic geometry by identifying solutions to an equation of two variables with points on a plane curve. To achieve geometric solutions without using coordinates, Bolzano introduced, in 1804, certain operations on points, lines and  planes, which are predecessors of vectors. introduced the notion of barycentric coordinates. introduced the notion of a bipoint, i.e., an oriented segment one of whose ends is the origin and the other one a target. Vectors were reconsidered with the presentation of complex numbers by Argand and Hamilton and the inception of quaternions by the latter. They are elements in R and R; treating them using linear combinations goes back to Laguerre in 1867, who also defined systems of linear equations. In 1857, Cayley introduced the matrix notation which allows for a harmonization and simplification of linear maps. Around the  same time, Grassmann studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations. In his work, the concepts of linear independence and dimension, as well as scalar products are present. Actually Grassmann's 1844 work exceeds the framework of vector spaces, since his considering multiplication, too, led him to what are today called algebras. Italian mathematician Peano was the first to give the modern definition of vector spaces and linear maps in 1888. An important development of vector spaces is due to the construction of function spaces by Henri Lebesgue. This was later formalized  by Banach and Hilbert, around 1920. At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of "p"-integrable functions and Hilbert spaces. Also at this time, the first studies concerning infinite-dimensional vector spaces were done. Examples. Arrows in the plane. The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, and , the parallelogram spanned by these two arrows contains one diagonal arrow that starts  at the origin, too. This new arrow is called the "sum" of the two arrows, and is denoted . 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 positive real number , the arrow that has the same direction as , but is dilated or shrunk by multiplying its length by , is called "multiplication" of by . It is denoted . When is  negative, is defined as the arrow pointing in the opposite direction instead. The following shows a few examples: if , the resulting vector has the same direction as , but is stretched to the double length of (right image below). Equivalently, is the sum . Moreover, has the opposite direction and the same length as (blue vector pointing down in the right image). Second example: ordered pairs of numbers. A second key example of a vector space is provided by pairs of real numbers and . (The order of the components and is significant, so such a pair is also called an ordered pair .) Such a pair is written as . The sum of two such pairs and multiplication of a pair with a number is defined as follows: and The first example above reduces to this example, if an arrow is represented by a pair of Cartesian coordinates of its endpoint. Coordinate space. The simplest example of a vector space over a field is the field itself (as it is an abelian group for addition, a part of requirements to be a field.), equipped with its addition (It becomes vector addition.) and multiplication (It becomes scalar multiplication.). More generally, all -tuples (sequences of length  ) of elements of form a vector space that is usually denoted and called a coordinate space. The case is the above-mentioned simplest example, in which the field is also regarded as a vector space over itself. The case and (so R) was discussed in the introduction above. Complex numbers and other field extensions. The set of complex numbers , that is, numbers that can be written in the form for real numbers and where is the imaginary unit, form a vector space over the reals with the usual addition and multiplication: and for real numbers , , , and . The various axioms of  a vector space follow from the fact that the same rules hold for complex number arithmetic. In fact, the example of complex numbers is essentially the same as (that is, it is "isomorphic" to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number as representing the ordered pair in the complex 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 and algebraic number theory: a  field containing a smaller field is an -vector space, by the given multiplication and addition operations of . For example, the complex numbers are a vector space over , and the field extension formula_22 is a vector space over . Function spaces. Functions from any fixed set to a field also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions and is the function given by and similarly for multiplication. Such function spaces occur in many geometric situations, when is the real line or an interval, or other subsets of . Many notions in topology and  analysis, such as continuity, integrability or differentiability are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property. Therefore, the set of such functions are vector spaces, whose study belongs to functional analysis. Linear equations. Systems of homogeneous linear equations are closely tied to vector spaces. For example, the solutions of are 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 formula_23 is the matrix containing the coefficients of the given equations, is the vector , denotes the matrix product, and is the zero vector. In a similar vein, the solutions of homogeneous "linear differential equations" form vector spaces. For example, yields , where and are arbitrary constants, and is the natural exponential function. Linear maps and matrices. The relation of two vector spaces can be expressed by "linear map" or "linear transformation". They are functions that reflect the vector space structure, that is, they preserve sums and scalar multiplication: An "isomorphism  is a linear map such that there exists an inverse map , which is a map such that the two possible compositions and are identity maps. Equivalently, is both one-to-one (injective) and onto (surjective). If there exists an isomorphism between and , the two spaces are said to be isomorphic"; they are then essentially identical as vector spaces, since all identities holding in are, via , transported to similar ones in , and vice versa via . For example, the "arrows in the plane" and "ordered pairs of numbers" vector spaces in the introduction are isomorphic: a planar arrow departing at the origin  of some (fixed) coordinate system can be expressed as an ordered pair by considering the - and -component of the arrow, as shown in the image at the right. Conversely, given a pair , the arrow going by to the right (or to the left, if is negative), and up (down, if is negative) turns back the arrow . Linear maps between two vector spaces form a vector space , also denoted , or . The space of linear maps from to is called the "dual vector space", denoted . Via the injective natural map , any vector space can be embedded into its "bidual"; the map is  an isomorphism if and only if the space is finite-dimensional. Once a basis of is chosen, linear maps are completely determined by specifying the images of the basis vectors, because any element of is expressed uniquely as a linear combination of them. If , a 1-to-1 correspondence between fixed bases of and gives rise to a linear map that maps any basis element of to the corresponding basis element of . It is an isomorphism, by its very definition. Therefore, two vector spaces are isomorphic if their dimensions agree and vice versa. Another way to express this is that any  vector space is "completely classified" (up to isomorphism) by its dimension, a single number. In particular, any "n"-dimensional -vector space is isomorphic to . There is, however, no "canonical" or preferred isomorphism; actually an isomorphism is equivalent to the choice of a basis of , by mapping the standard basis of to , via . The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context; see below. Matrices. "Matrices" are a useful notion to encode linear maps. They are written as a rectangular array of scalars as in the image at the right. Any -by- matrix gives rise to  a linear map from to , by the following or, using the matrix multiplication of the matrix with the coordinate vector : Moreover, after choosing bases of and , "any" linear map is uniquely represented by a matrix via this assignment. The determinant of a square matrix 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. The linear transformation of corresponding to a real "n"-by-"n" matrix is orientation preserving if and only if its determinant is positive. Eigenvalues and eigenvectors. Endomorphisms, linear maps , are particularly  important since in this case vectors can be compared with their image under , . Any nonzero vector satisfying , where is a scalar, is called an "eigenvector" of with "eigenvalue" . Equivalently, is an element of the kernel of the difference (where Id is the identity map . If is finite-dimensional, this can be rephrased using determinants: having eigenvalue is equivalent to By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in , called the characteristic polynomial of . If the field is large enough to contain a zero of  this polynomial (which automatically happens for algebraically closed, such as ) any linear map has at least one eigenvector. The vector space may or may not possess an eigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by the Jordan canonical form of the map. The set of all eigenvectors corresponding to a particular eigenvalue of forms a vector space known as the "eigenspace" corresponding to the eigenvalue (and ) in question. To achieve the spectral theorem, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see below. Basic constructions. In addition to the above  concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. In addition to the definitions given below, they are also characterized by universal properties, which determine an object by specifying the linear maps from to any other vector space. Subspaces and quotient spaces. A nonempty subset "W" of a vector space "V" that is closed under addition and scalar multiplication (and therefore contains the 0-vector of "V") is called a "linear subspace" of "V", or simply a "subspace" of "V", when the ambient space is unambiguously a vector space. Subspaces  of "V" are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set "S" of vectors is called its span, and it is the smallest subspace of "V" containing the set "S". Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of "S". A linear subspace of dimension 1 is a vector line. A linear subspace of dimension 2 is a vector plane. A linear subspace that contains all elements but one of a basis of the ambient space is a vector  hyperplane. In a vector space of finite dimension , a vector hyperplane is thus a subspace of dimension . The counterpart to subspaces are "quotient vector spaces". Given any subspace , the quotient space "V"/"W" (""V" modulo "W"") is defined as follows: as a set, it consists of where v is an arbitrary vector in "V". The sum of two such elements and is and scalar multiplication is given by . The key point in this definition is that if and only if the difference of v and v lies in "W". This way, the quotient space "forgets" information that is contained in the  subspace "W". The kernel ker("f") of a linear map consists of vectors v that are mapped to 0 in "W". The kernel and the image are subspaces of "V" and "W", respectively. The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field "F") is an abelian category, that is, a corpus of mathematical objects and structure-preserving maps between them (a category) that behaves much like the category of abelian groups. Because of this, many statements such as the first isomorphism theorem (also called rank–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 for groups. An important example is the kernel of a linear map for some fixed matrix "A", as above. The kernel of this map is the subspace of vectors x such that , which is precisely the set of solutions to the system of homogeneous linear equations belonging to "A". This concept also extends to linear differential equations In the corresponding map the derivatives of the function "f" appear linearly (as opposed to "f"′′("x"), for example). Since differentiation  is a linear procedure (that is, and for a constant ) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation form a vector space (over or ). Direct product and direct sum. The "direct product" of vector spaces and the "direct sum" of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space. The "direct product" formula_29 of a family of vector spaces "V" consists of the set of all tuples (, which specify for each index "i" in some index set "I" an element v of  V. Addition and scalar multiplication is performed componentwise. A variant of this construction is the "direct sum" formula_30 (also called coproduct and denoted formula_31), where only tuples with finitely many nonzero vectors are allowed. If the index set "I" is finite, the two constructions agree, but in general they are different. Tensor product. The "tensor product" , or simply , of two vector spaces "V" and "W" is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map is called bilinear if "g" is linear in both variables v and  w. That is to say, for fixed w the map is linear in the sense above and likewise for fixed v. The tensor product is a particular vector space that is a "universal" recipient of bilinear maps "g", as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called tensors subject to the rules These rules ensure that the map "f" from the to that maps a tuple to is bilinear. The universality states that given "any" vector space "X" and "any" bilinear map , there exists a unique map "u", shown in the diagram  with a dotted arrow, whose composition with "f" equals "g": . This is called the universal 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. Vector spaces with additional structure. From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces "per se" do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions converges to another  function. Likewise, linear algebra is not adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. Therefore, the needs of functional analysis require considering additional structures. A vector space may be given a partial order ≤, under which some vectors can be compared. For example, "n"-dimensional real space R can be ordered by comparing its vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions where formula_33 denotes the positive part of  formula_34 and formula_35 the negative part. Normed vector spaces and inner product spaces. "Measuring" vectors is done by specifying a norm, a datum which measures lengths of vectors, or by an inner product, which measures angles between vectors. Norms and inner products are denoted formula_36 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 as "normed vector spaces" and "inner product spaces", respectively. Coordinate space "F" can be equipped with the standard dot product: In R, this reflects the  common notion of the angle between two vectors x and y, by the law of cosines: Because of this, two vectors satisfying formula_39 are called orthogonal. An important variant of the standard dot product is used in Minkowski space: R endowed with the Lorentz product In contrast to the standard dot product, it is not positive definite: formula_41 also takes negative values, for example, for formula_42. Singling out the fourth coordinate—corresponding to time, as opposed to three space-dimensions—makes it useful for the mathematical treatment of special relativity. Topological vector spaces. Convergence questions are treated by considering vector  spaces "V" carrying a compatible topology, a structure that allows one to talk about elements being close to each other. Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if x and y in "V", and "a" in "F" vary by a bounded amount, then so do and . To make sense of specifying the amount a scalar changes, the field "F" also has to carry a topology in this context; a common choice are the reals or the complex numbers. In such "topological vector spaces" one can consider series of vectors. The infinite sum denotes  the limit of the corresponding finite partial sums of the sequence ("f") of elements of "V". For example, the "f" could be (real or complex) functions belonging to some function space "V", in which case the series is a function series. The mode of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are two prominent examples. A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any Cauchy sequence has a limit; such a vector space is called  complete. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval [0,1], equipped with the topology of uniform convergence is not complete because any continuous function on [0,1] can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem. In contrast, the space of "all" continuous functions on [0,1] with the same topology is complete. A norm gives rise to a topology by defining that a sequence of vectors v converges to v if and only if Banach 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 of functional analysis—focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence. The image at the right shows the equivalence of the 1-norm and ∞-norm on R: 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 called functionals) , maps between topological vector spaces are required to be continuous. In particular, the (topological) dual space consists of continuous functionals (or to ). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals. Banach spaces. "Banach spaces", introduced by Stefan Banach, are complete normed vector spaces. A first  example is the vector space formula_45 consisting of infinite vectors with real entries formula_46 whose formula_47-norm formula_48 given by The topologies on the infinite-dimensional space formula_45 are inequivalent for different formula_47. For example, the sequence of vectors formula_54, in which the first formula_55 components are formula_56 and the following ones are formula_57, converges to the zero vector for formula_58, but does not for formula_59: More generally than sequences of real numbers, functions formula_62 are endowed with a norm that replaces the above sum by the Lebesgue integral The space of integrable functions on a given domain formula_64 (for  example an interval) satisfying formula_65, and equipped with this norm are called Lebesgue spaces, denoted formula_66. These spaces are complete. (If one uses the Riemann integral instead, the space is "not" complete, which may be seen as a justification for Lebesgue's integration theory.) Concretely this means that for any sequence of Lebesgue-integrable functions   formula_67   with formula_68, satisfying the condition there exists a function formula_70 belonging to the vector space formula_66 such that Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces. Hilbert spaces. Complete inner product spaces are known as  Hilbert spaces, in honor of David Hilbert. The Hilbert space "L"(Ω), with inner product given by where formula_74 denotes the complex conjugate of "g"("x"), is a key case. By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions "f" with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the Taylor approximation, established an approximation of differentiable functions "f" by polynomials. By the Stone–Weierstrass theorem, every continuous function on can be approximated as closely as desired by a polynomial. A similar approximation  technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see below. 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 "H", in the sense that the "closure" of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a "basis" of "H", its cardinality is known as the Hilbert space dimension. Not only does the theorem exhibit suitable basis functions as sufficient  for approximation purposes, but also together with the Gram–Schmidt process, it enables one to construct a basis of orthogonal vectors. Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional Euclidean space. The solutions to various differential 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. As an example from physics, the time-dependent Schrödinger equation in quantum mechanics describes the change of physical properties in  time by means of a partial differential equation, whose solutions are called wavefunctions. Definite values for physical properties such as energy, or momentum, correspond to eigenvalues of a certain (linear) differential operator and the associated wavefunctions are called eigenstates. The spectral theorem decomposes a linear compact operator acting on functions in terms of these eigenfunctions and their eigenvalues. Algebras over fields. General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional bilinear operator defining the multiplication of two vectors is an "algebra over a field". Many algebras stem from functions on some geometrical  object: since functions with values in a given field can be multiplied pointwise, these entities form algebras. The Stone–Weierstrass theorem, for example, relies on Banach algebras which are both Banach spaces and algebras. Commutative algebra makes great use of rings of polynomials in one or several variables, introduced above. Their multiplication is both commutative and associative. These rings and their quotients form the basis of algebraic geometry, because they are rings of functions of algebraic geometric objects. Another crucial example are "Lie 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 "n"-by-"n" matrices, with , the commutator of two matrices, and , endowed with the cross product. The tensor algebra T("V") is a formal way of adding products to any vector space "V" to obtain an algebra. As a vector space, it is spanned by symbols, called simple tensors The multiplication is given by concatenating such symbols, imposing the distributive 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 above. In general , there are no relations between and . Forcing two such elements to be equal leads to the symmetric algebra, whereas forcing yields the exterior algebra. When a field, is explicitly stated, a common term used is -algebra. Related structures. Vector bundles. A "vector bundle" is a family of vector spaces parametrized continuously by a topological space "X". More precisely, a vector bundle over "X" is a topological space "E" equipped with a continuous map such that for every "x" in "X", the fiber π("x") is a vector space. The case dim is called a line bundle. For any vector space "V , the projection makes the product into a trivial" vector bundle. Vector bundles over "X" are required to be locally a product of "X" and some (fixed) vector space "V": for every "x" in "X", there is a neighborhood "U" of "x" such that the restriction of π to π("U") is isomorphic to the trivial bundle . Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space "X") be "twisted" in the large (that is, the bundle need not be (globally isomorphic to) the trivial bundle ). For example, the Möbius strip can be seen as a  line bundle over the circle "S" (by identifying open intervals with the real line). It is, however, different from the cylinder , because the latter is orientable whereas the former is not. Properties of certain vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle "S" is globally isomorphic to , since there is a global nonzero vector field on "S". In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2 -sphere "S" which is everywhere nonzero. K-theory studies the isomorphism classes of all vector bundles over some topological space. In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real division algebras: R, C, the quaternions H and the octonions O. The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space. Sections of that bundle are known as differential one-forms. Modules. "Modules" are to rings what vector spaces are to fields: the same axioms , applied to a ring "R" instead of a field "F", yield modules. The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have multiplicative inverses. For example, modules need not have bases, as the Z-module (that is, abelian group) Z/2Z shows; those modules that do (including all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a module over a ring which is a field, with the elements being called vectors. Some authors use the term "vector space" to mean modules  over a division ring. The algebro-geometric interpretation of commutative rings via their spectrum allows the development of concepts such as locally free modules, the algebraic counterpart to vector bundles. Affine and projective spaces. Roughly, "affine spaces" are vector spaces whose origins are not specified. More precisely, an affine space is a set with a free transitive vector space action. In particular, a vector space is an affine space over itself, by the map If "W" is a vector space, then an affine subspace is a subset of "W" obtained by translating a linear subspace "V" by a fixed vector  ; this space is denoted by (it is a coset of "V" in "W") and consists of all vectors of the form for An important example is the space of solutions of a system of inhomogeneous linear equations generalizing the homogeneous case above, which can be found by setting in this equation. The space of solutions is the affine subspace where x is a particular solution of the equation, and "V" is the space of solutions of the homogeneous equation (the nullspace of "A"). The set of one-dimensional subspaces of a fixed finite-dimensional vector space "V" is known as  projective space; it may be used to formalize the idea of parallel lines intersecting at infinity. Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension "k" and flags of subspaces, respectively.