Real coordinate space


In mathematics, the real coordinate space or real coordinate n-space, of dimension, denoted or, is the set of all ordered -tuples of real numbers, that is the set of all sequences of real numbers, also known as coordinate vectors.
Special cases are called the real line, the real coordinate plane, and the real coordinate three-dimensional space.
With component-wise addition and scalar multiplication, it is a real vector space.
The coordinates 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 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 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 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, 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:
These properties and structures of make it fundamental in almost all areas of mathematics and their application domains, such as statistics, probability theory, 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 . 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. If
  • is continuous
  • is continuous
then is not necessarily continuous. Continuity is a stronger condition: the continuity of in the natural topology, 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 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 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 and on elements of via right multiplication. The formula for left multiplication, a special case of matrix multiplication, is:
Any linear transformation is a continuous function. 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 of its matrix. If one permutes coordinates, the resulting orientation will depend on the parity of the permutation.
Diffeomorphisms of 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 has different properties depending on evenness of. For even it preserves orientation, while for odd it is reversed.

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 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, which allows only convex combinations.
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, a cone is an algebra over the universal orthant, and a convex set is an algebra over the universal simplex. This geometrizes the axioms in terms of "sums with 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 and sum of values in those points with the same coefficients.

Euclidean space

The dot product
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 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 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. 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.
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, and contact structure. All these structures, although can be defined in a coordinate-free manner, admit standard 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 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. 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, 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.