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: real coordinate space or complex coordinate space.

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.

In this article, vectors are represented in boldface to distinguish them from scalars.[nb 1]

A vector space over a field F is a set V together with two binary operations that satisfy the eight axioms listed below. In this context, the elements of V are commonly called vectors, and the elements of F are called scalars.

For having a vector space these two operations must satisfy the eight following axioms that must be satisfied for every u, v and w in V, and a and b in F.[1]

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 F are also commonly considered. Such a vector space is called an F-vector space or a vector space over F.

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 F into the endomorphism ring of this group.

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.

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:

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.[2] To achieve geometric solutions without using coordinates, Bolzano introduced, in 1804, certain operations on points, lines and planes, which are predecessors of vectors.[3] Möbius (1827) introduced the notion of barycentric coordinates. Bellavitis (1833) introduced the notion of a bipoint, i.e., an oriented segment one of whose ends is the origin and the other one a target.[4] Vectors were reconsidered with the presentation of complex numbers by Argand and Hamilton and the inception of quaternions by the latter.[5] They are elements in R2 and R4; 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.[6] 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.[7]

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.[8] 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.[9] Also at this time, the first studies concerning infinite-dimensional vector spaces were done.

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, v and w, 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 v + 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 positive real number a, the arrow that has the same direction as v, but is dilated or shrunk by multiplying its length by a, is called multiplication of v by a. It is denoted av. When a is negative, av is defined as the arrow pointing in the opposite direction instead.

The following shows a few examples: if a = 2, the resulting vector aw has the same direction as w, but is stretched to the double length of w (right image below). Equivalently, 2w is the sum w + w. Moreover, (−1)v = −v has the opposite direction and the same length as v (blue vector pointing down in the right image).

A second key example of a vector space is provided by pairs of real numbers x and y. (The order of the components x and y is significant, so such a pair is also called an ordered pair.) Such a pair is written as (x, y). The sum of two such pairs and multiplication of a pair with a number is defined as follows:

The first example above reduces to this example, if an arrow is represented by a pair of Cartesian coordinates of its endpoint.

The simplest example of a vector space over a field F is the field F 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 n-tuples (sequences of length n)

of elements ai of F form a vector space that is usually denoted Fn and called a coordinate space.[10] The case n = 1 is the above-mentioned simplest example, in which the field F is also regarded as a vector space over itself. The case F = R and n = 2 (so R2) was discussed in the introduction above.

The set of complex numbers C, that is, numbers that can be written in the form x + iy for real numbers x and y where i is the imaginary unit, form a vector space over the reals with the usual addition and multiplication: (x + iy) + (a + ib) = (x + a) + i(y + b) and c ⋅ (x + iy) = (cx) + i(cy) for real numbers x, y, a, b and c. 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 x + i y as representing the ordered pair (x, y) in the complex plane then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example.

Functions from any fixed set Ω to a field F also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions f and g is the function (f + g) given by

and similarly for multiplication. Such function spaces occur in many geometric situations, when Ω is the real line or an interval, or other subsets of R. 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.[12] Therefore, the set of such functions are vector spaces, whose study belongs to functional analysis.

Systems of homogeneous linear equations are closely tied to vector spaces.[13] For example, the solutions of

are given by triples with arbitrary a, b = a/2, and c = −5a/2. 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

yields f(x) = a ex + bx ex, where a and b are arbitrary constants, and ex is the natural exponential function.

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 f : VW such that there exists an inverse map g : WV, which is a map such that the two possible compositions fg : WW and gf : VV are identity maps. Equivalently, f is both one-to-one (injective) and onto (surjective).[15] 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, since all identities holding in V are, via f, transported to similar ones in W, and vice versa via g.

For example, the "arrows in the plane" and "ordered pairs of numbers" vector spaces in the introduction are isomorphic: a planar arrow v departing at the origin of some (fixed) coordinate system can be expressed as an ordered pair by considering the x- and y-component of the arrow, as shown in the image at the right. Conversely, given a pair (x, y), the arrow going by x to the right (or to the left, if x is negative), and y up (down, if y is negative) turns back the arrow v.

Linear maps VW between two vector spaces form a vector space HomF(V, W), also denoted L(V, W), or 𝓛(V, W).[16] The space of linear maps from V to F is called the dual vector space, denoted V.[17] Via the injective natural map VV∗∗, any vector space can be embedded into its bidual; the map is an isomorphism if and only if the space is finite-dimensional.[18]

Once a basis of V is chosen, linear maps f : VW are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them.[19] If dim V = dim W, a 1-to-1 correspondence between fixed bases of V and W gives rise to a linear map that maps any basis element of V to the corresponding basis element of W. It is an isomorphism, by its very definition.[20] 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 F-vector space V is isomorphic to Fn. There is, however, no "canonical" or preferred isomorphism; actually an isomorphism φ : FnV is equivalent to the choice of a basis of V, by mapping the standard basis of Fn to V, via φ. The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context; see below.[clarification needed]

Matrices are a useful notion to encode linear maps.[21] They are written as a rectangular array of scalars as in the image at the right. Any m-by-n matrix A gives rise to a linear map from Fn to Fm, by the following

or, using the matrix multiplication of the matrix A with the coordinate vector x:

Moreover, after choosing bases of V and W, any linear map f : VW is uniquely represented by a matrix via this assignment.[22]

The determinant det (A) of a square matrix A 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.[23] The linear transformation of Rn corresponding to a real n-by-n matrix is orientation preserving if and only if its determinant is positive.

Endomorphisms, linear maps f : VV, are particularly important since in this case vectors v can be compared with their image under f, f(v). Any nonzero vector v satisfying λv = f(v), where λ is a scalar, is called an eigenvector of f with eigenvalue λ.[nb 4][24] Equivalently, v is an element of the kernel of the difference fλ · Id (where Id is the identity map VV). If V is finite-dimensional, this can be rephrased using determinants: f having eigenvalue λ 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 λ, called the characteristic polynomial of f.[25] If the field F is large enough to contain a zero of this polynomial (which automatically happens for F algebraically closed, such as F = C) any linear map has at least one eigenvector. The vector space V may or may not possess an eigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by the Jordan canonical form of the map.[26][nb 5] The set of all eigenvectors corresponding to a particular eigenvalue of f forms a vector space known as the eigenspace corresponding to the eigenvalue (and f) in question. To achieve the spectral theorem, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see below.[clarification needed]

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 X by specifying the linear maps from X to any other vector space.

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.[27][nb 6] 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.[28]

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 n, a vector hyperplane is thus a subspace of dimension n – 1.

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 xAx for some fixed matrix A, as above.[clarification needed] The kernel of this map is the subspace of vectors x such that Ax = 0, 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

the derivatives of the function f appear linearly (as opposed to f′′(x)2, for example). Since differentiation is a linear procedure (that is, (f + g)′ = f′ + g and (c·f)′ = c·f for a constant c) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation D(f) = 0 form a vector space (over R or C).

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 tensor product VF W, or simply VW, 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 g : V × WX is called bilinear if g is linear in both variables v and w. That is to say, for fixed w the map vg(v, w) 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

Commutative diagram depicting the universal property of the tensor product.

These rules ensure that the map f from the V × W to VW that maps a tuple (v, w) to vw is bilinear. The universality states that given any vector space X and any bilinear map g : V × WX, there exists a unique map u, shown in the diagram with a dotted arrow, whose composition with f equals g: u(vw) = g(v, w).[35] 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.

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.[36] For example, n-dimensional real space Rn 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

In R2, this reflects the common notion of the angle between two vectors x and y, by the law of cosines:

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.[40][41] 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 x + y and ax.[nb 8] 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 (fi)iN of elements of V. For example, the fi 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.[42] In contrast, the space of all continuous functions on [0,1] with the same topology is complete.[43] A norm gives rise to a topology by defining that a sequence of vectors vn 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.[44] The image at the right shows the equivalence of the 1-norm and ∞-norm on R2: 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) VW, maps between topological vector spaces are required to be continuous.[45] In particular, the (topological) dual space V consists of continuous functionals VR (or to C). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.[46]

Banach spaces, introduced by Stefan Banach, are complete normed vector spaces.[47]

Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.[49]

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 as Hilbert spaces, in honor of David Hilbert.[50] The Hilbert space L2(Ω), with inner product given by

By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions fn 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.[52] By the Stone–Weierstrass theorem, every continuous function on [a, b] can be approximated as closely as desired by a polynomial.[53] A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see below.[clarification needed] 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.[nb 12] 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.[54] 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.[55] 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.[56] 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 [57]

spectral theorem decomposes a linear compact operator acting on functions in terms of these eigenfunctions and their eigenvalues.

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.[58] 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.[clarification needed] 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.[59]

Another crucial example are Lie algebras, which are neither commutative nor associative, but the failure to be so is limited by the constraints ([x, y] denotes the product of x and y):

Examples include the vector space of n-by-n matrices, with [x, y] = xyyx, the commutator of two matrices, and R3, 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.[61] 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.[clarification needed] In general, there are no relations between v1v2 and v2v1. Forcing two such elements to be equal leads to the symmetric algebra, whereas forcing v1v2 = − v2v1 yields the exterior algebra.[62]

When a field, F is explicitly stated, a common term used is F-algebra.

A vector bundle is a family of vector spaces parametrized continuously by a topological space X.[63] 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−1(x) is a vector space. The case dim V = 1 is called a line bundle. For any vector space V, the projection X × VX makes the product X × V 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−1(U) is isomorphic[nb 13] to the trivial bundle U × VU. 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 X × V). For example, the Möbius strip can be seen as a line bundle over the circle S1 (by identifying open intervals with the real line). It is, however, different from the cylinder S1 × R, because the latter is orientable whereas the former is not.[64]

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 S1 is globally isomorphic to S1 × R, since there is a global nonzero vector field on S1.[nb 14] In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2-sphere S2 which is everywhere nonzero.[65] K-theory studies the isomorphism classes of all vector bundles over some topological space.[66] 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 are to rings what vector spaces are to fields: the same axioms, applied to a ring R instead of a field F, yield modules.[67] 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.[68] 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.

Roughly, affine spaces are vector spaces whose origins are not specified.[69] 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 xW; this space is denoted by x + V (it is a coset of V in W) and consists of all vectors of the form x + v for vV. 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 b = 0 in this equation.[clarification needed][70] The space of solutions is the affine subspace x + V 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.[71] Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension k and flags of subspaces, respectively.