Real coordinate space

The Cartesian product structure of

R 2

on Cartesian plane of ordered pairs

(x, y)

. Blue lines denote coordinate axes, horizontal green lines are integer

y

, vertical cyan lines are integer

x

, brown-orange lines show half-integer

x

y

, magenta and its tint show multiples of one tenth (best seen under magnification)

In mathematics, real coordinate space of $n$ dimensions, written R^$n$ (/ɑːrˈɛn/ ar-EN) (also written $ℝ n$ with blackboard bold) is a coordinate space that allows several ( $n$ ) real variables to be treated as a single variable. With various numbers of dimensions (sometimes unspecified), $R n$ is used in many areas of pure and applied mathematics, as well as in physics. With component-wise addition and scalar multiplication, it is the prototypical real vector space and is a frequently used representation of Euclidean $n$ -space. Due to the latter fact, geometric metaphors are widely used for $R n$ , namely a plane for $R 2$ and three-dimensional space for $R 3$ .

Definition and uses

For any natural number $n$ , the set $R n$ consists of all $n$ -tuples of real numbers ( $R$ ). It is called (the) " $n$ -dimensional real space". Depending on its construction from $n$ instances of the set $R$ , it inherits some of the latter's structure, notably:

When defined as the direct sum of vector spaces, addition and scalar multiplication are defined on $R n$ : see below
$R n$ is a topological space: see below

An element of $R n$ is written

{\mathbf x}=(x_{1},x_{2},\ldots ,x_{n})

where each $x i$ is a real number.

For each $n$ there exists only one $R n$ , the real $n$ -space.^[1]

Purely mathematical uses of $R n$ can be roughly classified as follows, although these uses overlap. First, linear algebra studies its own properties under vector addition and linear transformations and uses it as a model of any $n$ -dimensional real vector space. Second, it is used in mathematical analysis to represent the domain of a function of $n$ real variables in a uniform way, as well as a space to which the graph of a real-valued function of $n - 1$ real variables is a subset. The third use parametrizes geometric points with elements of $R n$ ; it is common in analytic, differential and algebraic geometries.

$R n$ , together with supplemental structures on it, is also extensively used in mathematical physics, dynamical systems theory, mathematical statistics and probability theory.

In applied mathematics, numerical analysis, and so on, arrays, sequences, and other collections of numbers in applications can be seen as the use of $R n$ too.

The domain of a function of several variables

Main articles: Multivariable calculus and Real multivariable function

Any function $f (x 1, x 2, \dots , x n)$ of $n$ real variables can be considered as a function on $R n$ (that is, with $R n$ as its domain). The use of the real $n$ -space, instead of several variables considered separately, can simplify notation and suggest reasonable definitions. Consider, for $n = 2$ , a function composition of the following form:

F(t)=f(g_{1}(t),g_{2}(t)),

where functions $g 1$ and $g 2$ are continuous. If

\forall x 1 \in R : f (x 1, \cdot)

is continuous (by

x 2

)

\forall x 2 \in R : f (\cdot, x 2)

is continuous (by

x 1

)

then $F$ is not necessarily continuous. Continuity is a stronger condition: the continuity of $f$ in the natural $R 2$ topology (discussed below), also called multivariable continuity, which is sufficient for continuity of the composition $F$ .

Vector space

The coordinate space $R n$ forms an $n$ -dimensional vector space over the field of real numbers with the addition of the structure of linearity, and is often still denoted $R n$ . The operations on $R n$ as a vector space are typically defined by

{\mathbf x}+{\mathbf y}=(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n})

\alpha {\mathbf x}=(\alpha x_{1},\alpha x_{2},\ldots ,\alpha x_{n}).

The zero vector is given by

{\mathbf 0}=(0,0,\ldots ,0)

and the additive inverse of the vector $x$ is given by

-{\mathbf x}=(-x_{1},-x_{2},\ldots ,-x_{n}).

This structure is important because any $n$ -dimensional real vector space is isomorphic to the vector space $R n$ .

Matrix notation

Main article: Matrix (mathematics)

In standard matrix notation, each element of $R n$ is typically written as a column vector

{\mathbf x}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}

and sometimes as a row vector:

{\mathbf x}={\begin{bmatrix}x_{1}&x_{2}&\dots &x_{n}\end{bmatrix}}.

The coordinate space $R n$ may then be interpreted as the space of all $n \times 1$ column vectors, or all $1 \times n$ row vectors with the ordinary matrix operations of addition and scalar multiplication.

Linear transformations from $R n$ to $R m$ may then be written as $m \times n$ matrices which act on the elements of $R n$ via left multiplication (when the elements of $R n$ are column vectors) and on elements of $R m$ via right multiplication (when they are row vectors). The formula for left multiplication, a special case of matrix multiplication, is:

(A{{\mathbf x}})_{k}=\sum \limits _{{l=1}}^{n}A_{{kl}}x_{l}

Any linear transformation is a continuous function (see below). Also, a matrix define an open map from $R n$ to $R m$ if and only if the rank of the matrix equals to $m$ .

Standard basis

Main article: Standard basis

The coordinate space $R n$ comes with a standard basis:

{\begin{aligned}{\mathbf e}_{1}&=(1,0,\ldots ,0)\\{\mathbf e}_{2}&=(0,1,\ldots ,0)\\&{}\ \vdots \\{\mathbf e}_{n}&=(0,0,\ldots ,1)\end{aligned}}

To see that this is a basis, note that an arbitrary vector in $R n$ can be written uniquely in the form

{\mathbf x}=\sum _{{i=1}}^{n}x_{i}{\mathbf {e}}_{i}.

Geometric properties and uses

Orientation

The fact that real numbers, unlike many other fields, constitute an ordered field yields an orientation structure on $R n$ . Any full-rank linear map of $R n$ 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 $R n$ or domains in it, by their virtue to avoid zero Jacobian, 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 $R n$ has different properties depending on evenness of $n$ . For even $n$ it preserves orientation, while for odd $n$ it is reversed (see also improper rotation).

Affine space

For more details on this topic, see Affine space.

$R n$ understood as an affine space is the same space, where $R n$ 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 should go in an affine $n$ -space, because it can be translated anywhere.

Convexity

The n-simplex (see below) is the standard convex set, that maps to every polytope, and is the intersection of the standard

(n + 1)

affine hyperplane (standard affine space) and the standard

(n + 1)

orthant (standard cone).

For more details on this topic, see Convex analysis.

In a real vector space, such as $R n$ , 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, which allows only convex combinations (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 $R \infty$ 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 $R n$ to real numbers, which is defined through an inequality between its value on a convex combination of points and sum of values in those points with the same coefficients.

Euclidean space

Main articles: Euclidean space and Cartesian coordinate system

The dot product

{\mathbf {x}}\cdot {\mathbf {y}}=\sum _{{i=1}}^{n}x_{i}y_{i}=x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n}

defines the norm $| x | =$ √x⋅x on the vector space $R n$ . If every vector has its Euclidean norm, then for any pair of points the distance

d({\mathbf {x}},{\mathbf {y}})=\|{\mathbf {x}}-{\mathbf {y}}\|={\sqrt {\sum _{{i=1}}^{n}(x_{i}-y_{i})^{2}}}

is defined, providing a metric space structure on $R n$ in addition to its affine structure.

As for vector space structure, the dot product and Euclidean distance usually are assumed to exist in $R n$ without special explanations. However, the real $n$ -space and a Euclidean $n$ -space are distinct objects, strictly speaking. Any Euclidean $n$ -space has a coordinate system 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 $R n$ , but it is not the only possible one. Actually, any positive-definite quadratic form $q$ defines its own "distance" √q(x − y), but it is not very different from the Euclidean one in the sense that

\exists C_{1}>0,\ \exists C_{2}>0,\ \forall {\mathbf {x}},{\mathbf {y}}\in {\mathbb {R}}^{n}:C_{1}d({\mathbf {x}},{\mathbf {y}})\leq {\sqrt {q({\mathbf {x}}-{\mathbf {y}})}}\leq C_{2}d({\mathbf {x}},{\mathbf {y}}).

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 $R n$ , or its affine transformation, does not magnify distances more than by some fixed $C 2$ , and does not make distances smaller than $1 ∕ C 1$ times, a fixed finite number times smaller.

The aforementioned equivalence of metric functions remains valid if √q(x − y) is replaced with $M (x - y)$ , where $M$ 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 $R n$ is not especially different from the Euclidean metric, $R n$ is not always distinguished from a Euclidean $n$ -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 $R n$ , this choice is the most common, and almost exclusive one in differential geometry.

On the other hand, Whitney embedding theorems state that any real differentiable $m$ -dimensional manifold can be embedded into $R 2 m$ .

Other appearances

Other structures considered on $R n$ include the one of a pseudo-Euclidean space, symplectic structure (even $n$ ), and contact structure (odd $n$ ). All these structures, although can be defined in a coordinate-free manner, admit standard (and reasonably simple) forms in coordinates.

$R n$ is also a real vector subspace of $C n$ which is invariant to complex conjugation; see also complexification.

Polytopes in Rⁿ

See also: Linear programming and Convex polytope

There are three families of polytopes which have simple representations in $R n$ spaces, for any $n$ , and can be used to visualize any affine coordinate system in a real $n$ -space. Vertices of a hypercube have coordinates $(x 1, x 2, \dots , x n)$ where each $x k$ 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 −1 and 1. An $n$ -hypercube can be thought of as the Cartesian product of $n$ identical intervals (such as the unit interval $[0,1]$ ) on the real line. As an $n$ -dimensional subset it can be described with a system of $2 n$ inequalities:

\displaystyle {\begin{matrix}0\leq x_{1}\leq 1\\\vdots \\0\leq x_{n}\leq 1\end{matrix}}

(for

[0,1]

)

\displaystyle {\begin{matrix}|x_{1}|\leq 1\\\vdots \\|x_{n}|\leq 1\end{matrix}}

(for

[-1,1]

)

Each vertex of the cross-polytope has, for some $k$ , the $x k$ coordinate equal to ±1 and all other coordinates equal to 0 (such that it is the $k$ th standard basis vector up to sign). This is a dual polytope of hypercube. As an $n$ -dimensional subset it can be described with a single inequality which uses the absolute value operation:

\sum \limits _{{k=1}}^{n}|x_{k}|\leq 1\,,

but this can be expressed with a system of $2 n$ linear inequalities as well.

The third polytope with simply enumerable coordinates is the standard simplex, whose vertices are $n$ standard basis vectors and the origin $(0, 0, \dots , 0)$ . As an $n$ -dimensional subset it is described with a system of $n + 1$ linear inequalities:

{\begin{matrix}0\leq x_{1}\\\vdots \\0\leq x_{n}\\\sum \limits _{{k=1}}^{n}x_{k}\leq 1\end{matrix}}

Replacement of all "≤" with "<" gives interiors of these polytopes.

Topological properties

The topological structure of $R n$ (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, $R n$ is a linear topological space (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 $R n$ to itself which are not isometries, there can be many Euclidean structures on $R n$ 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 $R n$ onto itself, or its parts such as a Euclidean open ball or the interior of a hypercube).

$R n$ has the topological dimension $n$ . An important result on the topology of $R n$ , that is far from superficial, is Brouwer's invariance of domain. Any subset of $R n$ (with its subspace topology) that is homeomorphic to another open subset of $R n$ is itself open. An immediate consequence of this is that $R m$ is not homeomorphic to $R n$ if $m \neq n$ – 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 $R n$ . A continuous (although not smooth) space-filling curve (an image of $R 1$ ) is possible.

Examples

Empty column vector,
the only element of

R 0

R 1

n ≤ 1

Cases of $0 \leq n \leq 1$ do not offer anything new: $R 1$ is the real line, whereas $R 0$ (the space of empty vectors) is a singleton, understood as a zero vector space. However, it is useful to include these as trivial cases of theories which are appropriate for different $n$ .

n = 2

Both hypercube and cross-polytope in

R 2

are squares, but coordinates of vertices are arranged differently

For more details on this topic, see Two-dimensional space.

For more details on this topic, see Cartesian plane.

n = 3

Cube (the hypercube) and octahedron (the cross-polytope) of

R 3

. Coordinates are not shown

For more details on this topic, see Three-dimensional space.

n = 4

For more details on this topic, see Four-dimensional space.

$R 4$ can be imagined using the fact that 16 points $(x 1, x 2, x 3, x 4)$ , where each $x k$ is either 0 or 1, are vertices of a tesseract (pictured), the 4-hypercube (see above).

The first major use of $R 4$ is a spacetime model: three spatial coordinates plus one temporal. This is usually associated with theory of relativity, although four dimensions were continuously used for such models since Galilei. The choice of theory leads to different structure, though: in Galilean relativity the $t$ 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 $R 4$ with a curved metric for most practical purposes, though. Any of these structures does not provide a (positive) metric on $R 4$ .

Euclidean $R 4$ 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, $n = 4$ is the only case where $R n$ admits a non-standard differential structure: see exotic R⁴.

Footnotes

↑ Unlike many situations in mathematics where a certain object is unique up to isomorphism, $R n$ is unique in the strong sense: any of its elements is described explicitly with its $n$ real coordinates.

References

Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.
Munkres, James (1999). Topology. Prentice-Hall. ISBN 0-13-181629-2.

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