In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the real coordinate space of
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
, denoted ( ) or is the set of the
-tuples of
real number
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 ...
s, 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-dimensio ...
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 sig ...
over any
basis 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
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 mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
in geometry. This approach of geometry was introduced by
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
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
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
, 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 -
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
s of
real number
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 ...
s (). 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 (mathematics), variable to calculus with Function of several real variables, functions of several variables: the Differential calculus, di ...
, the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
of a
function of several real variables and the codomain of a real
vector valued function are
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s 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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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 alg ...
(sum of the term by term product of the components), it is an
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. 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 poin ...
, and a
topological vector space.
* It is a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
and a real
affine space, and every Euclidean or affine space is isomorphic to it.
* It is an
analytic manifold, and can be considered as the prototype of all
manifolds, 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
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
,
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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
.
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
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
). 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
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 ...
over the
field 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 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
and sometimes as a
row vector:
The coordinate space may then be interpreted as the space of all
column vectors, or all
row vectors 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, is:
Any linear transformation is a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
(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, unlike many other
fields, 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
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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 form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s, 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 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 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, which contains all ''non-negative'' linear combinations of its vectors. Corresponding concept in an affine space is a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
, which allows only
convex combinations (non-negative linear combinations that sum to 1).
In the language of
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study ...
, 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
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 dimension. ...
(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 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 alg ...
defines the
norm 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 setti ...
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 sig ...
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, 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 does not require that its model space should be , this choice is the most common, and almost exclusive one in
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
.
On the other hand,
Whitney embedding theorems 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 (even ), and
contact structure (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; see also
complexification.
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
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
) 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 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 of hypercube. As an -dimensional subset it can be described with a single inequality which uses the
absolute value 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
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* Open (Blues Image album), ''Open'' (Blues Image album), 1969
* Open (Gotthard album), ''Open'' (Gotthard album), 1999
* Open (C ...
in the Euclidean topology
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
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 .
An important result on the topology of , that is far from superficial, is
Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'.
Brouwer
* Adriaen Brouwer (1605–1638), Flemish painter
* Alexander Brouwer (b. 1989), Dutch beach volleyball player
* Andries Brou ...
'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
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, ...
(an image of ) is possible.
Examples
''n'' ≤ 1
Cases of do not offer anything new: is the
real line, whereas (the space containing the empty column vector) is a
singleton, 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 (pictured), the 4-hypercube (see
above).
The first major use of is a
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
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
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...
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
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 ...
. Some common examples are
* the
p-norm, defined by