In
mathematics, the real coordinate space of
dimension , denoted ( ) or is the set of the
-tuples of
real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a
real vector space, and its elements are called
coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...
s.
The
coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
over any
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
of the elements of a real vector space form a ''real coordinate space'' of the same dimension as that of the vector space. Similarly, the
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
of the points of a
Euclidean space of dimension form a ''real coordinate space'' of dimension .
These
one to one correspondences between vectors, points and coordinate vectors explain the names of ''coordinate space'' and ''coordinate vector''. It allows using
geometric terms and methods for studying real coordinate spaces, and, conversely, to use methods of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
in geometry. This approach of geometry was introduced by
René Descartes in the 17th century. It is widely used, as it allows locating points in Euclidean spaces, and computing with them.
Definition and structures
For any
natural number , the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
consists of all -
tuples of
real numbers (). It is called the "-dimensional real space" or the "real -space".
An element of is thus a -tuple, and is written
where each is a real number. So, in
multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather ...
, the
domain of a
function of several real variables and the codomain of a real
vector valued function are
subsets of for some .
The real -space has several further properties, notably:
* With
componentwise addition and
scalar multiplication, it is a
real vector space. Every -dimensional real vector space is
isomorphic to it.
* With the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
(sum of the term by term product of the components), it is an
inner product space. Every -dimensional real inner product space is isomorphic to it.
* As every inner product space, it is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, and a
topological vector space.
* It is a
Euclidean space and a real
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, and every Euclidean or affine space is isomorphic to it.
* It is an
analytic manifold
In mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic g ...
, and can be considered as the prototype of all
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s, as, by definition, a manifold is, near each point, isomorphic to an
open subset of .
* It is an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
, and every
real algebraic variety is a subset of .
These properties and structures of make it fundamental in almost all areas of mathematics and their application domains, such as
statistics,
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, and many parts of
physics.
The domain of a function of several variables
Any function of real variables can be considered as a function on (that is, with as its
domain). The use of the real -space, instead of several variables considered separately, can simplify notation and suggest reasonable definitions. Consider, for , a
function composition of the following form:
where functions and are
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
. If
* is continuous (by )
* is continuous (by )
then is not necessarily continuous. Continuity is a stronger condition: the continuity of in the natural topology (
discussed below), also called ''multivariable continuity'', which is sufficient for continuity of the composition .
Vector space
The coordinate space forms an -dimensional
vector space over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of real numbers with the addition of the structure of
linearity, and is often still denoted . The operations on as a vector space are typically defined by
The
zero vector is given by
and the
additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
of the vector is given by
This structure is important because any -dimensional real vector space is isomorphic to the vector space .
Matrix notation
In standard
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
notation, each element of is typically written as a
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
and sometimes as a
row vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
:
The coordinate space may then be interpreted as the space of all
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
s, or all
row vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
s with the ordinary matrix operations of addition and
scalar multiplication.
Linear transformations from to may then be written as matrices which act on the elements of via
left multiplication (when the elements of are column vectors) and on elements of via right multiplication (when they are row vectors). The formula for left multiplication, a special case of
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
, is:
Any linear transformation is a
continuous function (see
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
). Also, a matrix defines an
open map from to if and only if the
rank of the matrix equals to .
Standard basis
The coordinate space comes with a standard basis:
To see that this is a basis, note that an arbitrary vector in can be written uniquely in the form
Geometric properties and uses
Orientation
The fact that
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, unlike many other
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
, constitute an
ordered field yields an
orientation structure on . Any
full-rank linear map of to itself either preserves or reverses orientation of the space depending on the
sign of the
determinant of its matrix. If one
permutes coordinates (or, in other words, elements of the basis), the resulting orientation will depend on the
parity of the permutation.
Diffeomorphisms of or
domains in it, by their virtue to avoid zero
Jacobian
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to:
* Jacobian matrix and determinant
* Jacobian elliptic functions
* Jacobian variety
*Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähle ...
, are also classified to orientation-preserving and orientation-reversing. It has important consequences for the theory of
differential forms, whose applications include
electrodynamics.
Another manifestation of this structure is that the
point reflection in has different properties depending on
evenness of . For even it preserves orientation, while for odd it is reversed (see also
improper rotation).
Affine space
understood as an affine space is the same space, where as a vector space
acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
by
translations. Conversely, a vector has to be understood as a "
difference between two points", usually illustrated by a directed
line segment connecting two points. The distinction says that there is no
canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical exampl ...
choice of where the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
should go in an affine -space, because it can be translated anywhere.
Convexity
In a real vector space, such as , one can define a convex
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines conn ...
, which contains all ''non-negative'' linear combinations of its vectors. Corresponding concept in an affine space is a
convex set, which allows only
convex combination
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other wor ...
s (non-negative linear combinations that sum to 1).
In the language of
universal algebra, a vector space is an algebra over the universal vector space of finite sequences of coefficients, corresponding to finite sums of vectors, while an affine space is an algebra over the universal affine hyperplane in this space (of finite sequences summing to 1), a cone is an algebra over the universal
orthant (of finite sequences of nonnegative numbers), and a convex set is an algebra over the universal
simplex (of finite sequences of nonnegative numbers summing to 1). This geometrizes the axioms in terms of "sums with (possible) restrictions on the coordinates".
Another concept from convex analysis is a
convex function from to real numbers, which is defined through an
inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
between its value on a convex combination of
points
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Points ...
and sum of values in those points with the same coefficients.
Euclidean space
The
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
defines the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
on the vector space . If every vector has its
Euclidean norm, then for any pair of points the distance
is defined, providing a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
structure on in addition to its affine structure.
As for vector space structure, the dot product and Euclidean distance usually are assumed to exist in without special explanations. However, the real -space and a Euclidean -space are distinct objects, strictly speaking. Any Euclidean -space has a
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
where the dot product and Euclidean distance have the form shown above, called
''Cartesian''. But there are ''many'' Cartesian coordinate systems on a Euclidean space.
Conversely, the above formula for the Euclidean metric defines the ''standard'' Euclidean structure on , but it is not the only possible one. Actually, any
positive-definite quadratic form defines its own "distance" , but it is not very different from the Euclidean one in the sense that
Such a change of the metric preserves some of its properties, for example the property of being a
complete metric space.
This also implies that any full-rank linear transformation of , or its
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
, does not magnify distances more than by some fixed , and does not make distances smaller than times, a fixed finite number times smaller.
The aforementioned equivalence of metric functions remains valid if is replaced with , where is any convex positive
homogeneous function of degree 1, i.e. a
vector norm (see
Minkowski distance for useful examples). Because of this fact that any "natural" metric on is not especially different from the Euclidean metric, is not always distinguished from a Euclidean -space even in professional mathematical works.
In algebraic and differential geometry
Although the definition of a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
does not require that its model space should be , this choice is the most common, and almost exclusive one in
differential geometry.
On the other hand,
Whitney embedding theorem
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:
*The strong Whitney embedding theorem states that any smooth real -dimensional manifold (required also to be Hausdorff ...
s state that any real
differentiable -dimensional manifold can be
embedded into .
Other appearances
Other structures considered on include the one of a
pseudo-Euclidean space,
symplectic structure
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Ham ...
(even ), and
contact structure
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution ma ...
(odd ). All these structures, although can be defined in a coordinate-free manner, admit standard (and reasonably simple) forms in coordinates.
is also a real vector subspace of which is invariant to
complex conjugation
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an Imaginary number, imaginary part equal in magnitude but opposite in Sign (mathematics), sign. That is, (if a and b are real, then) the complex ...
; see also
complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
.
Polytopes in R''n''
There are three families of
polytopes which have simple representations in spaces, for any , and can be used to visualize any affine coordinate system in a real -space. Vertices of a
hypercube have coordinates where each takes on one of only two values, typically 0 or 1. However, any two numbers can be chosen instead of 0 and 1, for example and 1. An -hypercube can be thought of as the Cartesian product of identical
intervals (such as the
unit interval ) on the real line. As an -dimensional subset it can be described with a
system of inequalities:
for , and
for .
Each vertex of the
cross-polytope
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
has, for some , the coordinate equal to
±1 and all other coordinates equal to 0 (such that it is the th
standard basis vector up to
sign). This is a
dual polytope
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...
of hypercube. As an -dimensional subset it can be described with a single inequality which uses the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
operation:
but this can be expressed with a system of linear inequalities as well.
The third polytope with simply enumerable coordinates is the
standard simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimensi ...
, whose vertices are standard basis vectors and
the origin . As an -dimensional subset it is described with a system of linear inequalities:
Replacement of all "≤" with "<" gives interiors of these polytopes.
Topological properties
The
topological structure of (called standard topology, Euclidean topology, or usual topology) can be obtained not only
from Cartesian product. It is also identical to the
natural topology induced by
Euclidean metric discussed above: a set is
open in the Euclidean topology
if and only if it contains an
open ball around each of its points. Also, is a
linear topological space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
(see
continuity of linear maps above), and there is only one possible (non-trivial) topology compatible with its linear structure. As there are many open linear maps from to itself which are not
isometries, there can be many Euclidean structures on which correspond to the same topology. Actually, it does not depend much even on the linear structure: there are many non-linear
diffeomorphisms (and other homeomorphisms) of onto itself, or its parts such as a Euclidean open ball or
the interior of a hypercube).
has the
topological dimension
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
topologically invariant way.
Informal discussion
For ordinary Euclidean ...
.
An important result on the topology of , that is far from superficial, is
Brouwer's
invariance of domain. Any subset of (with its
subspace topology) that is
homeomorphic to another open subset of is itself open. An immediate consequence of this is that is not
homeomorphic to if – an intuitively "obvious" result which is nonetheless difficult to prove.
Despite the difference in topological dimension, and contrary to a naïve perception, it is possible to map a lesser-dimensional real space continuously and
surjectively onto . A continuous (although not smooth)
space-filling curve (an image of ) is possible.
Examples
''n'' ≤ 1
Cases of do not offer anything new: is the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
, whereas (the space containing the empty column vector) is a
singleton
Singleton may refer to:
Sciences, technology Mathematics
* Singleton (mathematics), a set with exactly one element
* Singleton field, used in conformal field theory Computing
* Singleton pattern, a design pattern that allows only one instance ...
, understood as a
zero vector space. However, it is useful to include these as
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
cases of theories that describe different .
''n'' = 2
''n'' = 3
''n'' = 4
can be imagined using the fact that points , where each is either 0 or 1, are vertices of a
tesseract
In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...
(pictured), the 4-hypercube (see
above).
The first major use of is a
spacetime model: three spatial coordinates plus one
temporal. This is usually associated with
theory of relativity, although four dimensions were used for such models since
Galilei. The choice of theory leads to different structure, though: in
Galilean relativity the coordinate is privileged, but in Einsteinian relativity it is not. Special relativity is set in
Minkowski space. General relativity uses curved spaces, which may be thought of as with a
curved metric for most practical purposes. None of these structures provide a (positive-definite)
metric on .
Euclidean also attracts the attention of mathematicians, for example due to its relation to
quaternions, a 4-dimensional
real algebra themselves. See
rotations in 4-dimensional Euclidean space for some information.
In differential geometry, is the only case where admits a non-standard
differential structure: see
exotic R4.
Norms on
One could define many norms on the
vector space . Some common examples are
* the
p-norm
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbak ...
, defined by