Real coordinate space

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The Cartesian product structure of R2 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 or y, magenta and its tint show multiples of one tenth (best seen under magnification)

In mathematics, real coordinate space of n dimensions, written Rn (/ɑrˈɛnˈ/ ar-EN) (R with superscript n, also written n with blackboard bold R) is a coordinate space that allows several (n) real variables to be treated as a single variable. With various numbers of dimensions (sometimes unspecified), Rn is used in many areas of pure and applied mathematics, as well as in physics. It is the prototypical real vector space and a frequently used representation of Euclidean n-space. Due to the latter fact, geometric metaphors are widely used for Rn, namely a plane for R2 and three-dimensional space for R3.

Definition and uses[edit]

For any natural number n, the set Rn consists of all n-tuples of real numbers (R). It is called (the) "n-dimensional real space". As it can be constructed as Cartesian product of n instances of the set R, it inherits some of its structure, notably:

An element of Rn is written

\mathbf x = (x_1, x_2, \ldots, x_n)

where each xi is a real number.

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

Purely mathematical uses of Rn can be roughly classified as follows, although these uses overlap. First, linear algebra studies its own properties under vector addition and linear transformations and use 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 Rn; it is common in analytic, differential and algebraic geometries.

Rn, 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 Rn too.

The domain of a function of several variables[edit]

Any function f(x1, x2, … , xn) of n real variables can be considered as a function on Rn (with this domain). The use of the real n-space, instead of several variables considered separately, can simplify a 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 g1 and g2 are continuous. If

x1 ∈ R : f(x1, ·) is continuous (by x2)
x2 ∈ R : f(·, x2) is continuous (by x1)

then F is not necessarily continuous. It is a stronger condition: the continuity of f in the natural R2 topology (discussed below), also called multivariable continuity, which is sufficient for continuity of the composition F.

Vector space[edit]

Rn forms an n-dimensional vector space over the field of real numbers. The operations on Rn are 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 Rn.

Matrix notation[edit]

Main article: Matrix (mathematics)

In standard matrix notation, each element of Rn 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 Rn may then be interpreted as the space of all n × 1 column vectors, or all 1 × n row vectors with the ordinary matrix operations of addition and scalar multiplication.

Linear transformations from Rn to Rm may then be written as m × n matrices which act on the elements of Rn via left multiplication (when the elements of Rn are column vectors) and on elements of Rm 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 Rn to Rm if and only if the rank of the matrix equals to m.

Standard basis[edit]

Main article: Standard basis

The coordinate space Rn comes with a standard basis:


\begin{align}
\mathbf e_1 & = (1, 0, \ldots, 0) \\
\mathbf e_2 & = (0, 1, \ldots, 0) \\
& {}\  \vdots \\
\mathbf e_n & = (0, 0, \ldots, 1)
\end{align}

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

\mathbf x = \sum_{i=1}^n x_i \mathbf{e}_i.

Geometric properties and uses[edit]

Orientation[edit]

The fact that real numbers, unlike many other fields, constitute an ordered field yields an orientation structure on Rn. Any full-rank linear map of Rn 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 Rn 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 Rn 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[edit]

For more details on this topic, see Affine space.

Rn understood as an affine space is the same space, where Rn 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[edit]

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 Rn, 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 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 Rn 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[edit]

The dot product

\mathbf{x}\cdot\mathbf{y} = \sum_{i=1}^n x_iy_i = x_1y_1+x_2y_2+\cdots+x_ny_n

defines the norm | x | =xx on the vector space Rn. 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 Rn in addition to its affine structure.

Like for vector space structure, the dot product and Euclidean distance usually are assumed existing on Rn without special explanations. Though, the real n-space and a Euclidean n-space are distinct objects, strictly speaking. Any Euclidean n-space has a coordinate system where 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 formula for Euclidean metric above defines the standard Euclidean structure on Rn, but it is not the only possible one. Actually, any positive-definite quadratic form q defines its own "distance" q(xy), but it is not very different from the Euclidean one in the sense that

\exist C_1 > 0,\ \exist C_2 > 0,\ \forall \mathbf{x}, \mathbf{y} \in \mathbb{R}^n:
 C_1 d(\mathbf{x}, \mathbf{y}) \le \sqrt{q(\mathbf{x} - \mathbf{y})} \le
 C_2 d(\mathbf{x}, \mathbf{y}).

Such change of the metric preserves some its properties, for example the property to be a complete metric space. This also implies that any full-rank linear transformation of Rn, or its affine transformation, does not magnify distances more than by some fixed C2, and does not make distances smaller than 1 ∕ C1 times, a fixed finite number times smaller.

Aforementioned equivalence of metric functions remains in place if q(xy) is replaced with M(xy), 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 Rn is not especially different from Euclidean, Rn is not always distinguished from a Euclidean n-space even in professional mathematical works.

In algebraic and differential geometry[edit]

Although the definition of a manifold does not require that its model space should be Rn, 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 R2m.

Other appearances[edit]

Other structures considered on Rn 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.

Rn is also a real vector subspace of Cn which is invariant to complex conjugation; see also complexification.

Polytopes in Rn[edit]

There are three families of polytopes which have simple representations in Rn spaces, for any n, and can be used to visualize any affine coordinate system in a real n-space. Vertices of hypercube have coordinates (x1, x2, … , xn) where each xk is either 0 or 1. Actually, 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 2n inequalities:

\displaystyle\begin{matrix}
0 \le x_1 \le 1 \\
\vdots \\
0 \le x_n \le 1
\end{matrix}
(for [0,1])     \displaystyle\begin{matrix}
|x_1| \le 1 \\
\vdots \\
|x_n| \le 1
\end{matrix}
(for [−1,1])


Each vertex of the cross-polytope has, for some k, the xk coordinate equal to ±1 and all other coordinates equal to 0 (such that it is the kth 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| \le 1\,,

but this can be expressed with a system of 2n 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, … , 0). As an n-dimensional subset it is described with a system of n + 1 linear inequalities:

\begin{matrix}
0 \le x_1 \\
\vdots \\
0 \le x_n \\
\sum\limits_{k=1}^n x_k \le 1
\end{matrix}

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

Topological properties[edit]

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

Rn has the topological dimension n. An important result on the topology of Rn, that is far from superficial, is Brouwer's invariance of domain. Any subset of Rn (with its subspace topology) that is homeomorphic to another open subset of Rn is itself open. An immediate consequence of this is that Rm is not homeomorphic to Rn if mn – 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 Rn. A continuous (although not smooth) space-filling curve (an image of R1) is possible.

Examples[edit]

Real 0-space.svg
Empty column vector,
the only element of R0
Real 1-space, orthoplex.svg
R1

n ≤ 1[edit]

Cases of 0 ≤ n ≤ 1 do not offer anything new: R1 is the real line, whereas R0 (the space of empty vectors) is a singleton, understood as zero vector space. Though, it is useful to include these "trivial" cases to theories which are appropriate for different n.

n = 2[edit]

Both hypercube and cross-polytope in R2 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.
See also: SL2(R)

n = 3[edit]

Cube (the hypercube) and octahedron (the cross-polytope) of R3. Coordinates are not shown
For more details on this topic, see Three-dimensional space.


n = 4[edit]

4-cube 3D.png
For more details on this topic, see Four-dimensional space.

R4 can be imagined using the fact that 16 points (x1, x2, x3, x4), where each xk is either 0 or 1, are vertices of a tesseract (pictured), the 4-hypercube (see above).

The first major use of R4 is a spacetime model: three spacial 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 R4 with a curved metric for most practical purposes, though. Any of these structures does not provide a (positive) metric on R4.

Euclidean R4 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 Rn admits a non-standard differential structure: see exotic R4.

Generalizations[edit]

See also[edit]

Footnotes[edit]

  1. ^ Unlike many situations in mathematics where certain object is unique up to isomorphism, Rn is unique in the strong sense: any its element is described explicitly with its n real coordinates.

References[edit]