Vector space

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https://en.wikipedia.org/wiki/Vector_space

 

See also: Feature space

Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.

In mathematics and physics, 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 of 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 matrices, 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 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 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 F is a non-empty 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.

• The first operation, called vector addition or simply addition assigns to any two vectors v and w in V a third vector in V which is commonly written as v + w, and called the sum of these two vectors. 
• The second operation, called scalar multiplication，assigns to any scalar a in F and any vector v in V another vector in V, which is denoted av.

To have a vector space, the eight following axioms must be satisfied for every u, v and w in V, and a and b in F.

Axiom	Meaning
Associativity of vector addition	u + (v + w) = (u + v) + w
Commutativity of vector addition	u + v = v + u
Identity element of vector addition	There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
Inverse elements of vector addition	For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0.
Compatibility of scalar multiplication with field multiplication	a(bv) = (ab)v
Identity element of scalar multiplication	1v = v, where 1 denotes the multiplicative identity in F.
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 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 (related to vector addition) say that a vector space is an abelian group under addition, and the four remaining axioms (related to the scalar multiplication), say that this operation defines a ring homomorphism from the field F into the endomorphism ring of this group.

Subtraction of two vectors can be defined as

$${\displaystyle \mathbf{v}-\mathbf{w}=\mathbf{v}+(-\mathbf{w}).}$$

Direct consequences of the axioms include that, for every ${\displaystyle s\in F}$ and ${\displaystyle \mathbf{v}\in V,}$ one has

• ${\displaystyle 0\mathbf{v}=\mathbf{0},}$
• ${\displaystyle s\mathbf{0}=\mathbf{0},}$
• ${\displaystyle (-1)\mathbf{v}=-\mathbf{v},}$
• ${\displaystyle s\mathbf{v}=\mathbf{0}}$ implies ${\displaystyle s=0}$ or ${\displaystyle \mathbf{v}=\mathbf{0}.}$

Even more concisely, a vector space is an ${\displaystyle F}$-module, where ${\displaystyle F}$ is a field.

Related concepts and properties

A vector v in R² (blue) expressed in terms of different bases: using the standard basis of R²: v = xe₁ + ye₂ (black), and using a different, non-orthogonal basis: v = f₁ + f₂ (red).

Linear combination

Given a set G of elements of a F-vector space V, a linear combination of elements of G is an element of V of the form

$${\displaystyle a_1{\mathbf{g}}_1+a_2{\mathbf{g}}_2+\cdots +a_k{\mathbf{g}}_k,}$$

where ${\displaystyle a_1,\dots ,a_k\in F}$ and ${\displaystyle {\mathbf{g}}_1,\dots ,{\mathbf{g}}_k\in G.}$ The scalars ${\displaystyle a_1,\dots ,a_k}$ are called the coefficients of the linear combination.

Linear independence

The elements of a subset G of a F-vector space V are said to be linearly independent if no element of G can be written as a linear combination of the other elements of G. Equivalently, they are linearly independent if two linear combinations of elements of G define the same element of V 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.

Linear subspace

A linear subspace or vector subspace W of a vector space V is a non-empty subset of V that is closed under vector addition and scalar multiplication; that is, the sum of two elements of W and the product of an element of V by a scalar belong to W. This implies that every linear combination of elements of W belongs to W. 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.
The closure property also implies that every intersection of linear subspaces is a linear subspace.

Linear span

Given a subset G of a vector space V, the linear span or simply the span of G is the smallest linear subspace of V that contains G, in the sense that it is the intersection of all linear subspaces that contain G. The span of G is also the set of all linear combinations of elements of G.
If W is the span of G, one says that G spans or generates W, and that G is a spanning set or a generating set of W.

Basis and dimension

A subset of a vector space is a basis if its elements are linearly independent and span the vector space. Every vector space has at least one basis, generally many (see Basis (linear algebra) § Proof that every vector space has a basis). Moreover, all bases of a vector space have the same cardinality, which is called the dimension of the vector space (see Dimension theorem for vector spaces). 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 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.

Consider a basis ${\displaystyle ({\mathbf{b}}_1,{\mathbf{b}}_2,\dots ,{\mathbf{b}}_n)}$ of a vector space V of dimension n over a field F. The definition of a basis implies that every ${\displaystyle \mathbf{v}\in V}$ may be written

$${\displaystyle \mathbf{v}=a_1{\mathbf{b}}_1+\cdots +a_n{\mathbf{b}}_n,}$$

with ${\displaystyle a_1,\dots ,a_n}$ in F, and that this decomposition is unique. The scalars ${\displaystyle a_1,\dots ,a_n}$ are called the coordinates of v on the basis. They are also said to be the coefficients of the decomposition of v on the basis. One also says that the n-tuple of the coordinates is the coordinate vector of v on the basis, since the set ${\displaystyle F^n}$ of the n-tuples of elements of F is a vector space for componentwise addition and scalar multiplication, whose dimension is n.

The one-to-one correspondence between vectors and their coordinate 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. If, in turn, 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 matrices.

In summary, finite-dimensional linear algebra may be expressed in three equivalent languages:

• Vector spaces, which provide concise and coordinate-free statements,
• Matrices, which are convenient for expressing concisely explicit computations,
• Systems of linear equations, which provide more elementary formulations.

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. Möbius (1827) introduced the notion of barycentric coordinates. Bellavitis (1833) introduced an equivalence relation on directed line segments that share the same length and direction which he called equipollence. A Euclidean vector is then an equivalence class of that relation.

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 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. 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, although he called them "linear systems".

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

Main article: Examples of vector spaces

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, 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).

Second example: ordered pairs of numbers

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:

$${\displaystyle (x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)}$$

and

$${\displaystyle a(x,y)=(ax,ay).}$$

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 F is the field F itself (as it is an abelian group for addition, a part of the 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)

$${\displaystyle (a_1,a_2,\dots ,a_n)}$$

of elements a_i of F form a vector space that is usually denoted Fⁿ and called a coordinate space. 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.

Complex numbers and other field extensions

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) = (c ⋅ x) + i(c ⋅ y) 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.

More generally, field extensions provide another class of examples of vector spaces, particularly in algebra and algebraic number theory: a field F containing a smaller field E is an E-vector space, by the given multiplication and addition operations of F. For example, the complex numbers are a vector space over R, and the field extension ${\displaystyle \mathbf{Q}(i\sqrt{5})}$ is a vector space over Q.

Function spaces

Main article: Function space

Addition of functions: the sum of the sine and the exponential function is ${\displaystyle \sin +\exp :\mathbb{R}\to \mathbb{R}}$ with ${\displaystyle (\sin +\exp )(x)=\sin (x)+\exp (x)}$.

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 ${\displaystyle (f+g)}$ given by

$${\displaystyle (f+g)(w)=f(w)+g(w),}$$

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. Therefore, the set of such functions are vector spaces, whose study belongs to functional analysis.

Linear equations

Main articles: Linear equation, Linear differential equation, and Systems of linear equations

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

$${\displaystyle \begin{array}{rlrlrlrlr}& & a& & +3b& +& & c& =0\\ 4& & a& & +2b& +& 2& c& =0\end{array}}$$

are given by triples with arbitrary ${\displaystyle a,}$ ${\displaystyle b=a/2,}$ and ${\displaystyle c=-5a/2.}$ 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

$${\displaystyle A\mathbf{x}=\mathbf{0},}$$

where ${\displaystyle A=\left[\begin{array}{ccc}1& 3& 1\\ 4& 2& 2\end{array}\right]}$ is the matrix containing the coefficients of the given equations, ${\displaystyle \mathbf{x}}$ is the vector ${\displaystyle (a,b,c),}$ ${\displaystyle A\mathbf{x}}$ denotes the matrix product, and ${\displaystyle \mathbf{0}=(0,0)}$ is the zero vector. In a similar vein, the solutions of homogeneous linear differential equations form vector spaces. For example,

$${\displaystyle f^{\mathrm{\prime }\mathrm{\prime }}(x)+2f^{\mathrm{\prime }}(x)+f(x)=0}$$

yields ${\displaystyle f(x)=a{e}^{-x}+bx{e}^{-x},}$ where ${\displaystyle a}$ and ${\displaystyle b}$ are arbitrary constants, and ${\displaystyle e^x}$ is the natural exponential function.

Linear maps and matrices

Main article: Linear map

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:

$${\displaystyle f(\mathbf{v}+\mathbf{w})=f(\mathbf{v})+f(\mathbf{w}),\text{ and }}$$

$${\displaystyle f(a\cdot \mathbf{v})=a\cdot f(\mathbf{v})}$$

for all ${\displaystyle \mathbf{v}}$ and ${\displaystyle \mathbf{w}}$ in ${\displaystyle V,}$ all ${\displaystyle a}$ in ${\displaystyle F.}$

An isomorphism is a linear map f : V → W such that there exists an inverse map g : W → V, which is a map such that the two possible compositions f ∘ g : W → W and g ∘ f : V → V are identity maps. Equivalently, f is both one-to-one (injective) and onto (surjective). 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.

Describing an arrow vector v by its coordinates x and y yields an isomorphism of vector spaces.

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 V → W between two vector spaces form a vector space Hom_F(V, W), also denoted L(V, W), or 𝓛(V, W). The space of linear maps from V to F is called the dual vector space, denoted V^∗. Via the injective natural map V → V^∗∗, 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 V is chosen, linear maps f : V → W 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. 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. 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 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 : Fⁿ → V 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.

Matrices

Main articles: Matrix and Determinant

A typical matrix

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 m-by-n matrix ${\displaystyle A}$ gives rise to a linear map from Fⁿ to F^m, by the following

$${\displaystyle \mathbf{x}=(x_1,x_2,\dots ,x_n)\mapsto \left(\sum _{j=1}^n{a}_{1j}{x}_j,\sum _{j=1}^n{a}_{2j}{x}_j,\dots ,\sum _{j=1}^n{a}_{mj}{x}_j\right),}$$

where ${\displaystyle \sum }$ denotes summation, or, using the matrix multiplication of the matrix ${\displaystyle A}$ with the coordinate vector ${\displaystyle \mathbf{x}:}$

$${\displaystyle \mathbf{x}\mapsto A\mathbf{x}.}$$

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

The volume of this parallelepiped is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors r₁, r₂, and r₃.

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. The linear transformation of Rⁿ corresponding to a real n-by-n matrix is orientation preserving if and only if its determinant is positive.

Eigenvalues and eigenvectors

Main article: Eigenvalues and eigenvectors

Endomorphisms, linear maps f : V → V, 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. Equivalently, v is an element of the kernel of the difference f − λ · Id (where Id is the identity map V → V). If V is finite-dimensional, this can be rephrased using determinants: f having eigenvalue λ is equivalent to

$${\displaystyle det(f-\lambda \cdot \mathrm{Id})=0.}$$

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. 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. 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.

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 ${\displaystyle X}$ by specifying the linear maps from ${\displaystyle X}$ to any other vector space.

Subspaces and quotient spaces

Main articles: Linear subspace and Quotient vector space

A line passing through the origin (blue, thick) in R³ is a linear subspace. It is the intersection of two planes (green and yellow).

A nonempty subset ${\displaystyle W}$ of a vector space ${\displaystyle V}$ that is closed under addition and scalar multiplication (and therefore contains the ${\displaystyle \mathbf{0}}$-vector of ${\displaystyle V}$) is called a linear subspace of ${\displaystyle V,}$ or simply a subspace of ${\displaystyle V,}$ when the ambient space is unambiguously a vector space. Subspaces of ${\displaystyle V}$ are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set ${\displaystyle S}$ of vectors is called its span, and it is the smallest subspace of ${\displaystyle V}$ containing the set ${\displaystyle S.}$Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of ${\displaystyle S.}$

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 ${\displaystyle n,}$ a vector hyperplane is thus a subspace of dimension ${\displaystyle n-1.}$

The counterpart to subspaces are quotient vector spaces. Given any subspace ${\displaystyle W\subseteq V,}$ the quotient space ${\displaystyle V/W}$ ("${\displaystyle V}$ modulo ${\displaystyle W}$") is defined as follows: as a set, it consists of ${\displaystyle \mathbf{v}+W=\{\mathbf{v}+\mathbf{w}:\mathbf{w}\in W\},}$ where ${\displaystyle \mathbf{v}}$ is an arbitrary vector in ${\displaystyle V.}$ The sum of two such elements ${\displaystyle {\mathbf{v}}_1+W}$ and ${\displaystyle {\mathbf{v}}_2+W}$ is ${\displaystyle \left({\mathbf{v}}_1+{\mathbf{v}}_2\right)+W,}$ and scalar multiplication is given by ${\displaystyle a\cdot (\mathbf{v}+W)=(a\cdot \mathbf{v})+W.}$ The key point in this definition is that ${\displaystyle {\mathbf{v}}_1+W={\mathbf{v}}_2+W}$ if and only if the difference of ${\displaystyle {\mathbf{v}}_1}$ and ${\displaystyle {\mathbf{v}}_2}$ lies in ${\displaystyle W.}$ This way, the quotient space "forgets" information that is contained in the subspace ${\displaystyle W.}$

The kernel ${\displaystyle \ker (f)}$ of a linear map ${\displaystyle f:V\to W}$ consists of vectors ${\displaystyle \mathbf{v}}$ that are mapped to ${\displaystyle \mathbf{0}}$ in ${\displaystyle W.}$ The kernel and the image ${\displaystyle \mathrm{im}(f)=\{f(\mathbf{v}):\mathbf{v}\in V\}}$ are subspaces of ${\displaystyle V}$ and ${\displaystyle W,}$ respectively. The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field ${\displaystyle F}$) 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)

$${\displaystyle V/\ker (f)\equiv \mathrm{im}(f)}$$

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 ${\displaystyle \mathbf{x}\mapsto A\mathbf{x}}$ for some fixed matrix ${\displaystyle A,}$ as above. The kernel of this map is the subspace of vectors ${\displaystyle \mathbf{x}}$ such that ${\displaystyle A\mathbf{x}=\mathbf{0},}$ which is precisely the set of solutions to the system of homogeneous linear equations belonging to ${\displaystyle A.}$ This concept also extends to linear differential equations

$${\displaystyle a_{0}f+a_1\frac{df}{dx}+a_2\frac{d^{2}f}{d{x}^2}+\cdots +a_n\frac{d^{n}f}{d{x}^n}=0,}$$

where the coefficients ${\displaystyle a_i}$ are functions in ${\displaystyle x,}$ too. In the corresponding map

$${\displaystyle f\mapsto D(f)=\sum _{i=0}^n{a}_i\frac{d^{i}f}{d{x}^i},}$$

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

Direct product and direct sum

Main articles: Direct product and Direct sum of modules

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 ${\displaystyle \prod _{i\in I}{V}_i}$ of a family of vector spaces ${\displaystyle V_i}$ consists of the set of all tuples ${\displaystyle {\left({\mathbf{v}}_i\right)}_{i\in I},}$ which specify for each index ${\displaystyle i}$ in some index set ${\displaystyle I}$ an element ${\displaystyle {\mathbf{v}}_i}$ of ${\displaystyle V_i.}$ Addition and scalar multiplication is performed componentwise. A variant of this construction is the direct sum $\underset{i\in I}{⨁}{V}_i$ (also called coproduct and denoted $\coprod _{i\in I}{V}_i$), where only tuples with finitely many nonzero vectors are allowed. If the index set ${\displaystyle I}$ is finite, the two constructions agree, but in general they are different.

Tensor product

Main article: Tensor product of vector spaces

The tensor product ${\displaystyle V{\otimes }_{F}W,}$ or simply ${\displaystyle V\otimes W,}$ of two vector spaces ${\displaystyle V}$ and ${\displaystyle W}$ is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map ${\displaystyle g:V\times W\to X}$ from the Cartesian product ${\displaystyle V\times W}$ is called bilinear if ${\displaystyle g}$ is linear in both variables ${\displaystyle \mathbf{v}}$ and ${\displaystyle \mathbf{w}.}$ That is to say, for fixed ${\displaystyle \mathbf{w}}$ the map ${\displaystyle \mathbf{v}\mapsto g(\mathbf{v},\mathbf{w})}$ is linear in the sense above and likewise for fixed ${\displaystyle \mathbf{v}.}$

Commutative diagram depicting the universal property of the tensor product

The tensor product is a particular vector space that is a universal recipient of bilinear maps ${\displaystyle g,}$ as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called tensors

$${\displaystyle {\mathbf{v}}_1\otimes {\mathbf{w}}_1+{\mathbf{v}}_2\otimes {\mathbf{w}}_2+\cdots +{\mathbf{v}}_n\otimes {\mathbf{w}}_n,}$$

subject to the rules

$${\displaystyle \begin{array}{rlrl}a\cdot (\mathbf{v}\otimes \mathbf{w})& =(a\cdot \mathbf{v})\otimes \mathbf{w}=\mathbf{v}\otimes (a\cdot \mathbf{w}),& & \text{ where }a\text{ is a scalar}\\ ({\mathbf{v}}_1+{\mathbf{v}}_2)\otimes \mathbf{w}& ={\mathbf{v}}_1\otimes \mathbf{w}+{\mathbf{v}}_2\otimes \mathbf{w}& & \\ \mathbf{v}\otimes ({\mathbf{w}}_1+{\mathbf{w}}_2)& =\mathbf{v}\otimes {\mathbf{w}}_1+\mathbf{v}\otimes {\mathbf{w}}_2.& & \end{array}}$$

These rules ensure that the map ${\displaystyle f}$ from the ${\displaystyle V\times W}$ to ${\displaystyle V\otimes W}$ that maps a tuple ${\displaystyle (\mathbf{v},\mathbf{w})}$ to ${\displaystyle \mathbf{v}\otimes \mathbf{w}}$ is bilinear. The universality states that given any vector space ${\displaystyle X}$ and any bilinear map ${\displaystyle g:V\times W\to X,}$ there exists a unique map ${\displaystyle u,}$ shown in the diagram with a dotted arrow, whose composition with ${\displaystyle f}$ equals ${\displaystyle g:}$ ${\displaystyle u(\mathbf{v}\otimes \mathbf{w})=g(\mathbf{v},\mathbf{w}).}$ 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 over a given field 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 ${\displaystyle \le ,}$ under which some vectors can be compared. For example, ${\displaystyle n}$-dimensional real space ${\displaystyle {\mathbf{R}}^n}$ 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

$${\displaystyle f=f^{+}-f^{-}.}$$

where ${\displaystyle f^{+}}$ denotes the positive part of ${\displaystyle f}$ and ${\displaystyle f^{-}}$ the negative part.

Normed vector spaces and inner product spaces

Main articles: Normed vector space and Inner product space

"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 ${\displaystyle |\mathbf{v}|}$ and ${\displaystyle ⟨\mathbf{v},\mathbf{w}⟩,}$ respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm $|\mathbf{v}|:=\sqrt{⟨\mathbf{v},\mathbf{v}⟩}.$ Vector spaces endowed with such data are known as normed vector spaces and inner product spaces, respectively.

Coordinate space ${\displaystyle F^n}$ can be equipped with the standard dot product:

$${\displaystyle ⟨\mathbf{x},\mathbf{y}⟩=\mathbf{x}\cdot \mathbf{y}=x_1{y}_1+\cdots +x_n{y}_n.}$$

In ${\displaystyle {\mathbf{R}}^2,}$ this reflects the common notion of the angle between two vectors ${\displaystyle \mathbf{x}}$ and ${\displaystyle \mathbf{y},}$ by the law of cosines:

$${\displaystyle \mathbf{x}\cdot \mathbf{y}=\cos \left(\mathrm{\angle }(\mathbf{x},\mathbf{y})\right)\cdot |\mathbf{x}|\cdot |\mathbf{y}|.}$$

Because of this, two vectors satisfying ${\displaystyle ⟨\mathbf{x},\mathbf{y}⟩=0}$ are called orthogonal. An important variant of the standard dot product is used in Minkowski space: ${\displaystyle {\mathbf{R}}^4}$ endowed with the Lorentz product

$${\displaystyle ⟨\mathbf{x}|\mathbf{y}⟩=x_1{y}_1+x_2{y}_2+x_3{y}_3-x_4{y}_4.}$$

In contrast to the standard dot product, it is not positive definite: ${\displaystyle ⟨\mathbf{x}|\mathbf{x}⟩}$ also takes negative values, for example, for ${\displaystyle \mathbf{x}=(0,0,0,1).}$ 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

Main article: Topological vector space

Convergence questions are treated by considering vector spaces ${\displaystyle V}$ 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 ${\displaystyle \mathbf{x}}$ and ${\displaystyle \mathbf{y}}$ in ${\displaystyle V,}$ and ${\displaystyle a}$ in ${\displaystyle F}$ vary by a bounded amount, then so do ${\displaystyle \mathbf{x}+\mathbf{y}}$ and ${\displaystyle a\mathbf{x}.}$ To make sense of specifying the amount a scalar changes, the field ${\displaystyle F}$ 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

$${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{f}_i=\underset{n\to \mathrm{\infty }}{lim}{f}_1+\cdots +f_n}$$

denotes the limit of the corresponding finite partial sums of the sequence ${\displaystyle f_1,f_2,\dots }$ of elements of ${\displaystyle V.}$ For example, the ${\displaystyle f_i}$ could be (real or complex) functions belonging to some function space ${\displaystyle V,}$ 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.

Unit "spheres" in ${\displaystyle {\mathbf{R}}^2}$ consist of plane vectors of norm 1. Depicted are the unit spheres in different ${\displaystyle p}$-norms, for ${\displaystyle p=1,2,}$ and ${\displaystyle \mathrm{\infty }.}$ 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 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 ${\displaystyle [0,1],}$ equipped with the topology of uniform convergence is not complete because any continuous function on ${\displaystyle [0,1]}$ can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem. In contrast, the space of all continuous functions on ${\displaystyle [0,1]}$ with the same topology is complete. A norm gives rise to a topology by defining that a sequence of vectors ${\displaystyle {\mathbf{v}}_n}$ converges to ${\displaystyle \mathbf{v}}$ if and only if

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}|{\mathbf{v}}_n-\mathbf{v}|=0.}$$

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 ${\displaystyle 1}$-norm and ${\displaystyle \mathrm{\infty }}$-norm on ${\displaystyle {\mathbf{R}}^2:}$ 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) ${\displaystyle V\to W,}$ maps between topological vector spaces are required to be continuous. In particular, the (topological) dual space ${\displaystyle V^{\ast }}$ consists of continuous functionals ${\displaystyle V\to \mathbf{R}}$ (or to ${\displaystyle \mathbf{C}}$). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.

Banach spaces

Main article: Banach space

Banach spaces, introduced by Stefan Banach, are complete normed vector spaces.

A first example is the vector space ${\displaystyle {\ell }^p}$ consisting of infinite vectors with real entries ${\displaystyle \mathbf{x}=\left(x_1,x_2,\dots ,x_n,\dots \right)}$ whose ${\displaystyle p}$-norm ${\displaystyle (1\le p\le \mathrm{\infty })}$ given by

$${\displaystyle \Vert \mathbf{x}{\Vert }_{\mathrm{\infty }}:=\underset{i}{sup}|x_i|\text{ for }p=\mathrm{\infty },\text{ and }}$$

$${\displaystyle \Vert \mathbf{x}{\Vert }_p:={\left(\sum _i|x_i{|}^p\right)}^{\frac{1}{p}}\text{ for }p<\mathrm{\infty }.}$$

The topologies on the infinite-dimensional space ${\displaystyle {\ell }^p}$ are inequivalent for different ${\displaystyle p.}$ For example, the sequence of vectors ${\displaystyle {\mathbf{x}}_n=\left(2^{-n},2^{-n},\dots ,2^{-n},0,0,\dots \right),}$ in which the first ${\displaystyle 2^n}$ components are ${\displaystyle 2^{-n}}$ and the following ones are ${\displaystyle 0,}$ converges to the zero vector for ${\displaystyle p=\mathrm{\infty },}$ but does not for ${\displaystyle p=1:}$

$${\displaystyle \Vert {\mathbf{x}}_n{\Vert }_{\mathrm{\infty }}=sup(2^{-n},0)=2^{-n}\to 0,}$$

but

$${\displaystyle \Vert {\mathbf{x}}_n{\Vert }_1=\sum _{i=1}^{2^n}{2}^{-n}=2^n\cdot 2^{-n}=1.}$$

More generally than sequences of real numbers, functions ${\displaystyle f:\mathrm{\Omega }\to \mathbb{R}}$ are endowed with a norm that replaces the above sum by the Lebesgue integral

$${\displaystyle \Vert f{\Vert }_p:={\left({\int }_{\mathrm{\Omega }}|f(x){|}^{p}d\mu (x)\right)}^{\frac{1}{p}}.}$$

The space of integrable functions on a given domain ${\displaystyle \mathrm{\Omega }}$ (for example an interval) satisfying ${\displaystyle \Vert f{\Vert }_p<\mathrm{\infty },}$ and equipped with this norm are called Lebesgue spaces, denoted ${\displaystyle L^p(\mathrm{\Omega }).}$

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 ${\displaystyle f_1,f_2,\dots ,f_n,\dots }$ with ${\displaystyle \Vert f_n{\Vert }_p<\mathrm{\infty },}$ satisfying the condition

$${\displaystyle \underset{k,n\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}{\left|f_k(x)-f_n(x)\right|}^{p}d\mu (x)=0}$$

there exists a function ${\displaystyle f(x)}$ belonging to the vector space ${\displaystyle L^p(\mathrm{\Omega })}$ such that

$${\displaystyle \underset{k\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}{\left|f(x)-f_k(x)\right|}^{p}d\mu (x)=0.}$$

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

Hilbert spaces

Main article: Hilbert space

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. The Hilbert space ${\displaystyle L^2(\mathrm{\Omega }),}$ with inner product given by

$${\displaystyle ⟨f,g⟩={\int }_{\mathrm{\Omega }}f(x)\overline{g(x)}dx,}$$

where ${\displaystyle \overline{g(x)}}$ denotes the complex conjugate of ${\displaystyle g(x),}$ is a key case.

By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions ${\displaystyle f_n}$ 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 ${\displaystyle f}$ by polynomials. By the Stone–Weierstrass theorem, every continuous function on ${\displaystyle [a,b]}$ 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 ${\displaystyle H,}$ 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 ${\displaystyle H,}$ 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

Main articles: Algebra over a field and Lie algebra

A hyperbola, given by the equation ${\displaystyle x\cdot y=1.}$ The coordinate ring of functions on this hyperbola is given by ${\displaystyle \mathbf{R}[x,y]/(x\cdot y-1),}$ an infinite-dimensional vector space over ${\displaystyle \mathbf{R}.}$

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 (${\displaystyle [x,y]}$ denotes the product of ${\displaystyle x}$ and ${\displaystyle y}$):

• ${\displaystyle [x,y]=-[y,x]}$ (anticommutativity), and
• ${\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0}$ (Jacobi identity).

Examples include the vector space of ${\displaystyle n}$-by-${\displaystyle n}$ matrices, with ${\displaystyle [x,y]=xy-yx,}$ the commutator of two matrices, and ${\displaystyle {\mathbf{R}}^3,}$ endowed with the cross product.

The tensor algebra ${\displaystyle \mathrm{T}(V)}$ is a formal way of adding products to any vector space ${\displaystyle V}$ to obtain an algebra. As a vector space, it is spanned by symbols, called simple tensors

$${\displaystyle {\mathbf{v}}_1\otimes {\mathbf{v}}_2\otimes \cdots \otimes {\mathbf{v}}_n,}$$

where the degree ${\displaystyle n}$ varies. 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 ${\displaystyle {\mathbf{v}}_1\otimes {\mathbf{v}}_2}$ and ${\displaystyle {\mathbf{v}}_2\otimes {\mathbf{v}}_1.}$ Forcing two such elements to be equal leads to the symmetric algebra, whereas forcing ${\displaystyle {\mathbf{v}}_1\otimes {\mathbf{v}}_2=-{\mathbf{v}}_2\otimes {\mathbf{v}}_1}$ yields the exterior algebra.

When a field, ${\displaystyle F}$ is explicitly stated, a common term used is ${\displaystyle F}$-algebra.

Related structures

Vector bundles

Main articles: Vector bundle and Tangent bundle

A Möbius strip. Locally, it looks like U × R.

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

$${\displaystyle \pi :E\to X}$$

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 × V → X 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 to the trivial bundle U × V → U. 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.

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. In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2-sphere S² 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

Main article: Module

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

Main articles: Affine space and Projective space

An affine plane (light blue) in R³. It is a two-dimensional subspace shifted by a vector x (red).

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

$${\displaystyle V\times V\to W,(\mathbf{v},\mathbf{a})\mapsto \mathbf{a}+\mathbf{v}.}$$

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 x ∈ W; 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 v ∈ V. An important example is the space of solutions of a system of inhomogeneous linear equations

$${\displaystyle A\mathbf{v}=\mathbf{b}}$$

generalizing the homogeneous case above, which can be found by setting ${\displaystyle \mathbf{b}=\mathbf{0}}$ in this equation. 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. Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension k and flags of subspaces, respectively.

Related concepts

Specific vectors in a vector space

• Zero vector (sometimes also called null vector and denoted by ${\displaystyle \mathbf{0}}$), the additive identity in a vector space. In a normed vector space, it is the unique vector of norm zero. In a Euclidean vector space, it is the unique vector of length zero.
• Basis vector, an element of a given basis of a vector space.
• Unit vector, a vector in a normed vector space whose norm is 1, or a Euclidean vector of length one.
• Isotropic vector or null vector, in a vector space with a quadratic form, a non-zero vector for which the form is zero. If a null vector exists, the quadratic form is said an isotropic quadratic form.

Vectors in specific vector spaces

• Column vector, a matrix with only one column. The column vectors with a fixed number of rows form a vector space.
• Row vector, a matrix with only one row. The row vectors with a fixed number of columns form a vector space.
• Coordinate vector, the n-tuple of the coordinates of a vector on a basis of n elements. For a vector space over a field F, these n-tuples form the vector space ${\displaystyle F^n}$ (where the operation are pointwise addition and scalar multiplication).
• Displacement vector, a vector that specifies the change in position of a point relative to a previous position. Displacement vectors belong to the vector space of translations.
• Position vector of a point, the displacement vector from a reference point (called the origin) to the point. A position vector represents the position of a point in a Euclidean space or an affine space.
• Velocity vector, the derivative, with respect to time, of the position vector. It does not depend of the choice of the origin, and, thus belongs to the vector space of translations.
• Pseudovector, also called axial vector
• Covector, an element of the dual of a vector space. In an inner product space, the inner product defines an isomorphism between the space and its dual, which may make difficult to distinguish a covector from a vector. The distinction becomes apparent when one changes coordinates (non-orthogonally).
• Tangent vector, an element of the tangent space of a curve, a surface or, more generally, a differential manifold at a given point (these tangent spaces are naturally endowed with a structure of vector space)
• Normal vector or simply normal, in a Euclidean space or, more generally, in an inner product space, a vector that is perpendicular to a tangent space at a point.
• Gradient, the coordinates vector of the partial derivatives of a function of several real variables. In a Euclidean space the gradient gives the magnitude and direction of maximum increase of a scalar field. The gradient is a covector that is normal to a level curve.
• Four-vector, in the theory of relativity, a vector in a four-dimensional real vector space called Minkowski space

Exact solutions in general relativity

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Exact_solutions_in_general_relativity 

In general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter. Mathematically, finding an exact solution means finding a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field.

Background and definition

These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations). Following a standard recipe which is widely used in mathematical physics, these tensor fields should also give rise to specific contributions to the stress–energy tensor ${\displaystyle T^{\alpha \beta }}$. (A field is described by a Lagrangian, varying with respect to the field should give the field equations and varying with respect to the metric should give the stress-energy contribution due to the field.)

Finally, when all the contributions to the stress–energy tensor are added up, the result must be a solution of the Einstein field equations

${\displaystyle G^{\alpha \beta }=\kappa T^{\alpha \beta }.}$

In the above field equations, ${\displaystyle G^{\alpha \beta }}$ is the Einstein tensor, computed uniquely from the metric tensor which is part of the definition of a Lorentzian manifold. Since giving the Einstein tensor does not fully determine the Riemann tensor, but leaves the Weyl tensor unspecified (see the Ricci decomposition), the Einstein equation may be considered a kind of compatibility condition: the spacetime geometry must be consistent with the amount and motion of any matter or non-gravitational fields, in the sense that the immediate presence "here and now" of non-gravitational energy–momentum causes a proportional amount of Ricci curvature "here and now". Moreover, taking covariant derivatives of the field equations and applying the Bianchi identities, it is found that a suitably varying amount/motion of non-gravitational energy–momentum can cause ripples in curvature to propagate as gravitational radiation, even across vacuum regions, which contain no matter or non-gravitational fields.

Difficulties with the definition

Any Lorentzian manifold is a solution of the Einstein field equation for some right hand side. This is illustrated by the following procedure:

• take any Lorentzian manifold, compute its Einstein tensor ${\displaystyle G^{\alpha \beta }}$, which is a purely mathematical operation
• divide by the Einstein gravitational constant ${\displaystyle \kappa }$
• declare the resulting symmetric second rank tensor field to be the stress–energy tensor ${\displaystyle T^{\alpha \beta }}$.

This shows that there are two complementary ways to use general relativity:

• One can fix the form of the stress–energy tensor (from some physical reasons, say) and study the solutions of the Einstein equations with such right hand side (for example, if the stress–energy tensor is chosen to be that of the perfect fluid, a spherically symmetric solution can serve as a stellar model)
• Alternatively, one can fix some geometrical properties of a spacetime and look for a matter source that could provide these properties. This is what cosmologists have done since the 2000s: they assume that the Universe is homogeneous, isotropic, and accelerating and try to realize what matter (called dark energy) can support such a structure.

Within the first approach the alleged stress–energy tensor must arise in the standard way from a "reasonable" matter distribution or non-gravitational field. In practice, this notion is pretty clear, especially if we restrict the admissible non-gravitational fields to the only one known in 1916, the electromagnetic field. But ideally we would like to have some mathematical characterization that states some purely mathematical test which we can apply to any putative "stress–energy tensor", which passes everything which might arise from a "reasonable" physical scenario, and rejects everything else. No such characterization is known. Instead, we have crude tests known as the energy conditions, which are similar to placing restrictions on the eigenvalues and eigenvectors of a linear operator. On the one hand, these conditions are far too permissive: they would admit "solutions" which almost no-one believes are physically reasonable. On the other, they may be far too restrictive: the most popular energy conditions are apparently violated by the Casimir effect.

Einstein also recognized another element of the definition of an exact solution: it should be a Lorentzian manifold (meeting additional criteria), i.e. a smooth manifold. But in working with general relativity, it turns out to be very useful to admit solutions which are not everywhere smooth; examples include many solutions created by matching a perfect fluid interior solution to a vacuum exterior solution, and impulsive plane waves. Once again, the creative tension between elegance and convenience, respectively, has proven difficult to resolve satisfactorily.

In addition to such local objections, we have the far more challenging problem that there are very many exact solutions which are locally unobjectionable, but globally exhibit causally suspect features such as closed timelike curves or structures with points of separation ("trouser worlds"). Some of the best known exact solutions, in fact, have globally a strange character.

Types of exact solution

Many well-known exact solutions belong to one of several types, depending upon the intended physical interpretation of the stress–energy tensor:

• Vacuum solutions: ${\displaystyle T^{\alpha \beta }=0}$; these describe regions in which no matter or non-gravitational fields are present,
• Electrovacuum solutions: ${\displaystyle T^{\alpha \beta }}$ must arise entirely from an electromagnetic field which solves the source-free Maxwell equations on the given curved Lorentzian manifold; this means that the only source for the gravitational field is the field energy (and momentum) of the electromagnetic field,
• Null dust solutions: ${\displaystyle T^{\alpha \beta }}$ must correspond to a stress–energy tensor which can be interpreted as arising from incoherent electromagnetic radiation, without necessarily solving the Maxwell field equations on the given Lorentzian manifold,
• Fluid solutions: ${\displaystyle T^{\alpha \beta }}$ must arise entirely from the stress–energy tensor of a fluid (often taken to be a perfect fluid); the only source for the gravitational field is the energy, momentum, and stress (pressure and shear stress) of the matter comprising the fluid.

In addition to such well established phenomena as fluids or electromagnetic waves, one can contemplate models in which the gravitational field is produced entirely by the field energy of various exotic hypothetical fields:

• Scalar field solutions: ${\displaystyle T^{\alpha \beta }}$ must arise entirely from a scalar field (often a massless scalar field); these can arise in classical field theory treatments of meson beams, or as quintessence,
• Lambdavacuum solutions (not a standard term, but a standard concept for which no name yet exists): ${\displaystyle T^{\alpha \beta }}$ arises entirely from a nonzero cosmological constant.

One possibility which has received little attention (perhaps because the mathematics is so challenging) is the problem of modeling an elastic solid. Presently, it seems that no exact solutions for this specific type are known.

Below we have sketched a classification by physical interpretation. Solutions can also be organized using the Segre classification of the possible algebraic symmetries of the Ricci tensor:

• non-null electrovacuums have Segre type ${\displaystyle \{(1,1)(11)\}}$ and isotropy group SO(1,1) x SO(2),
• null electrovacuums and null dusts have Segre type ${\displaystyle \{(2,11)\}}$ and isotropy group E(2),
• perfect fluids have Segre type ${\displaystyle \{1,(111)\}}$ and isotropy group SO(3),
• Lambda vacuums have Segre type ${\displaystyle \{(1,111)\}}$ and isotropy group SO(1,3).

The remaining Segre types have no particular physical interpretation and most of them cannot correspond to any known type of contribution to the stress–energy tensor.

Examples

Noteworthy examples of vacuum solutions, electrovacuum solutions, and so forth, are listed in specialized articles (see below). These solutions contain at most one contribution to the energy–momentum tensor, due to a specific kind of matter or field. However, there are some notable exact solutions which contain two or three contributions, including:

• NUT-Kerr–Newman–de Sitter solution contains contributions from an electromagnetic field and a positive vacuum energy, as well as a kind of vacuum perturbation of the Kerr vacuum which is specified by the so-called NUT parameter,
• Gödel dust contains contributions from a pressureless perfect fluid (dust) and from a positive vacuum energy.

Constructing solutions

The Einstein field equations are a system of coupled, nonlinear partial differential equations. In general, this makes them hard to solve. Nonetheless, several effective techniques for obtaining exact solutions have been established.

The simplest involves imposing symmetry conditions on the metric tensor, such as stationarity (symmetry under time translation) or axisymmetry (symmetry under rotation about some symmetry axis). With sufficiently clever assumptions of this sort, it is often possible to reduce the Einstein field equation to a much simpler system of equations, even a single partial differential equation (as happens in the case of stationary axisymmetric vacuum solutions, which are characterized by the Ernst equation) or a system of ordinary differential equations (as happens in the case of the Schwarzschild vacuum).

This naive approach usually works best if one uses a frame field rather than a coordinate basis.

A related idea involves imposing algebraic symmetry conditions on the Weyl tensor, Ricci tensor, or Riemann tensor. These are often stated in terms of the Petrov classification of the possible symmetries of the Weyl tensor, or the Segre classification of the possible symmetries of the Ricci tensor. As will be apparent from the discussion above, such Ansätze often do have some physical content, although this might not be apparent from their mathematical form.

This second kind of symmetry approach has often been used with the Newman–Penrose formalism, which uses spinorial quantities for more efficient bookkeeping.

Even after such symmetry reductions, the reduced system of equations is often difficult to solve. For example, the Ernst equation is a nonlinear partial differential equation somewhat resembling the nonlinear Schrödinger equation (NLS).

But recall that the conformal group on Minkowski spacetime is the symmetry group of the Maxwell equations. Recall too that solutions of the heat equation can be found by assuming a scaling Ansatz. These notions are merely special cases of Sophus Lie's notion of the point symmetry of a differential equation (or system of equations), and as Lie showed, this can provide an avenue of attack upon any differential equation which has a nontrivial symmetry group. Indeed, both the Ernst equation and the NLS have nontrivial symmetry groups, and some solutions can be found by taking advantage of their symmetries. These symmetry groups are often infinite dimensional, but this is not always a useful feature.

Emmy Noether showed that a slight but profound generalization of Lie's notion of symmetry can result in an even more powerful method of attack. This turns out to be closely related to the discovery that some equations, which are said to be completely integrable, enjoy an infinite sequence of conservation laws. Quite remarkably, both the Ernst equation (which arises several ways in the studies of exact solutions) and the NLS turn out to be completely integrable. They are therefore susceptible to solution by techniques resembling the inverse scattering transform which was originally developed to solve the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation which arises in the theory of solitons, and which is also completely integrable. Unfortunately, the solutions obtained by these methods are often not as nice as one would like. For example, in a manner analogous to the way that one obtains a multiple soliton solution of the KdV from the single soliton solution (which can be found from Lie's notion of point symmetry), one can obtain a multiple Kerr object solution, but unfortunately, this has some features which make it physically implausible.

There are also various transformations (see Belinski-Zakharov transform) which can transform (for example) a vacuum solution found by other means into a new vacuum solution, or into an electrovacuum solution, or a fluid solution. These are analogous to the Bäcklund transformations known from the theory of certain partial differential equations, including some famous examples of soliton equations. This is no coincidence, since this phenomenon is also related to the notions of Noether and Lie regarding symmetry. Unfortunately, even when applied to a "well understood", globally admissible solution, these transformations often yield a solution which is poorly understood and their general interpretation is still unknown.

Existence of solutions

Given the difficulty of constructing explicit small families of solutions, much less presenting something like a "general" solution to the Einstein field equation, or even a "general" solution to the vacuum field equation, a very reasonable approach is to try to find qualitative properties which hold for all solutions, or at least for all vacuum solutions. One of the most basic questions one can ask is: do solutions exist, and if so, how many?

To get started, we should adopt a suitable initial value formulation of the field equation, which gives two new systems of equations, one giving a constraint on the initial data, and the other giving a procedure for evolving this initial data into a solution. Then, one can prove that solutions exist at least locally, using ideas not terribly dissimilar from those encountered in studying other differential equations.

To get some idea of "how many" solutions we might optimistically expect, we can appeal to Einstein's constraint counting method. A typical conclusion from this style of argument is that a generic vacuum solution to the Einstein field equation can be specified by giving four arbitrary functions of three variables and six arbitrary functions of two variables. These functions specify initial data, from which a unique vacuum solution can be evolved. (In contrast, the Ernst vacuums, the family of all stationary axisymmetric vacuum solutions, are specified by giving just two functions of two variables, which are not even arbitrary, but must satisfy a system of two coupled nonlinear partial differential equations. This may give some idea of how just tiny a typical "large" family of exact solutions really is, in the grand scheme of things.)

However, this crude analysis falls far short of the much more difficult question of global existence of solutions. The global existence results which are known so far turn out to involve another idea.

Global stability theorems

We can imagine "disturbing" the gravitational field outside some isolated massive object by "sending in some radiation from infinity". We can ask: what happens as the incoming radiation interacts with the ambient field? In the approach of classical perturbation theory, we can start with Minkowski vacuum (or another very simple solution, such as the de Sitter lambdavacuum), introduce very small metric perturbations, and retain only terms up to some order in a suitable perturbation expansion—somewhat like evaluating a kind of Taylor series for the geometry of our spacetime. This approach is essentially the idea behind the post-Newtonian approximations used in constructing models of a gravitating system such as a binary pulsar. However, perturbation expansions are generally not reliable for questions of long-term existence and stability, in the case of nonlinear equations.

The full field equation is highly nonlinear, so we really want to prove that the Minkowski vacuum is stable under small perturbations which are treated using the fully nonlinear field equation. This requires the introduction of many new ideas. The desired result, sometimes expressed by the slogan that the Minkowski vacuum is nonlinearly stable, was finally proven by Demetrios Christodoulou and Sergiu Klainerman only in 1993. Analogous results are known for lambdavac perturbations of the de Sitter lambdavacuum (Helmut Friedrich) and for electrovacuum perturbations of the Minkowski vacuum (Nina Zipser). In contrast, anti-de Sitter spacetime is known to be unstable under certain conditions.

The positive energy theorem

Main article: Positive energy theorem

Another issue we might worry about is whether the net mass-energy of an isolated concentration of positive mass-energy density (and momentum) always yields a well-defined (and non-negative) net mass. This result, known as the positive energy theorem was finally proven by Richard Schoen and Shing-Tung Yau in 1979, who made an additional technical assumption about the nature of the stress–energy tensor. The original proof is very difficult; Edward Witten soon presented a much shorter "physicist's proof", which has been justified by mathematicians—using further very difficult arguments. Roger Penrose and others have also offered alternative arguments for variants of the original positive energy theorem.