Real coordinate space
Template:Short description Template:No footnotes
In mathematics, the real coordinate space or real coordinate n-space, of dimension Template:Mvar, denoted Template:Math or Template:Nowrap, is the set of all ordered [[tuple|Template:Mvar-tuples]] of real numbers, that is the set of all sequences of Template:Mvar real numbers, also known as coordinate vectors. Special cases are called the real line Template:Math, the real coordinate plane Template:Math, and the real coordinate three-dimensional space Template:Math. 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 Template:Mvar, Template:Math (Euclidean line, Template:Math; Euclidean plane, Template:Math; Euclidean three-dimensional space, Template:Math) form a real coordinate space of dimension Template:Mvar.
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 structuresEdit
For any natural number Template:Mvar, the set Template:Math consists of all Template:Mvar-tuples of real numbers (Template:Math). It is called the "Template:Mvar-dimensional real space" or the "real Template:Mvar-space".
An element of Template:Math is thus a Template:Mvar-tuple, and is written <math display="block">(x_1, x_2, \ldots, x_n)</math> where each Template:Math 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 Template:Math for some Template:Mvar.
The real Template:Mvar-space has several further properties, notably:
- With componentwise addition and scalar multiplication, it is a real vector space. Every Template:Mvar-dimensional real vector space is isomorphic to it.
- With the dot product (sum of the term by term product of the components), it is an inner product space. Every Template:Mvar-dimensional real inner product space is isomorphic to it.
- As every inner product space, it is a topological space, and a topological vector space.
- It is a Euclidean space 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 Template:Math.
- It is an algebraic variety, and every real algebraic variety is a subset of Template:Math.
These properties and structures of Template:Math 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 variablesEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Any function Template:Math of Template:Mvar real variables can be considered as a function on Template:Math (that is, with Template:Math as its domain). The use of the real Template:Mvar-space, instead of several variables considered separately, can simplify notation and suggest reasonable definitions. Consider, for Template:Math, a function composition of the following form: <math display="block"> F(t) = f(g_1(t),g_2(t)),</math> where functions Template:Math and Template:Math are continuous. If
- Template:Math is continuous (by Template:Math)
- Template:Math is continuous (by Template:Math)
then Template:Mvar is not necessarily continuous. Continuity is a stronger condition: the continuity of Template:Mvar in the natural Template:Math topology (discussed below), also called multivariable continuity, which is sufficient for continuity of the composition Template:Mvar.
Vector spaceEdit
The coordinate space Template:Math forms an Template:Mvar-dimensional vector space over the field of real numbers with the addition of the structure of linearity, and is often still denoted Template:Math. The operations on Template:Math as a vector space are typically defined by <math display="block">\mathbf x + \mathbf y = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n)</math> <math display="block">\alpha \mathbf x = (\alpha x_1, \alpha x_2, \ldots, \alpha x_n).</math> The zero vector is given by <math display="block">\mathbf 0 = (0, 0, \ldots, 0)</math> and the additive inverse of the vector Template:Math is given by <math display="block">-\mathbf x = (-x_1, -x_2, \ldots, -x_n).</math>
This structure is important because any Template:Mvar-dimensional real vector space is isomorphic to the vector space Template:Math.
Matrix notationEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In standard matrix notation, each element of Template:Math is typically written as a column vector <math display="block">\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}</math> and sometimes as a row vector: <math display="block">\mathbf x = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix}.</math>
The coordinate space Template:Math may then be interpreted as the space of all Template:Math column vectors, or all Template:Math row vectors with the ordinary matrix operations of addition and scalar multiplication.
Linear transformations from Template:Math to Template:Math may then be written as Template:Math matrices which act on the elements of Template:Math via left multiplication (when the elements of Template:Math are column vectors) and on elements of Template:Math via right multiplication (when they are row vectors). The formula for left multiplication, a special case of matrix multiplication, is: <math display="block">(A{\mathbf x})_k = \sum_{l=1}^n A_{kl} x_l</math>
Template:AnchorAny linear transformation is a continuous function (see below). Also, a matrix defines an open map from Template:Math to Template:Math if and only if the rank of the matrix equals to Template:Mvar.
Standard basisEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The coordinate space Template:Math comes with a standard basis: <math display="block">\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}</math>
To see that this is a basis, note that an arbitrary vector in Template:Math can be written uniquely in the form <math display="block">\mathbf x = \sum_{i=1}^n x_i \mathbf{e}_i.</math>
Geometric properties and usesEdit
OrientationEdit
The fact that real numbers, unlike many other fields, constitute an ordered field yields an orientation structure on Template:Math. Any full-rank linear map of Template:Math 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 Template:Math 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 Template:Math has different properties depending on [[even and odd numbers|evenness of Template:Mvar]]. For even Template:Mvar it preserves orientation, while for odd Template:Mvar it is reversed (see also improper rotation).
Affine spaceEdit
Template:Details Template:Math understood as an affine space is the same space, where Template:Math 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 Template:Mvar-space, because it can be translated anywhere.
ConvexityEdit
Template:Details In a real vector space, such as Template:Math, 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 Template:Math 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 Template:Math 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 spaceEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The dot product <math display="block">\mathbf{x}\cdot\mathbf{y} = \sum_{i=1}^n x_iy_i = x_1y_1+x_2y_2+\cdots+x_ny_n</math> defines the norm Template:Math on the vector space Template:Math. If every vector has its Euclidean norm, then for any pair of points the distance <math display="block">d(\mathbf{x}, \mathbf{y}) = \|\mathbf{x} - \mathbf{y}\| = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}</math> is defined, providing a metric space structure on Template:Math in addition to its affine structure.
As for vector space structure, the dot product and Euclidean distance usually are assumed to exist in Template:Math without special explanations. However, the real Template:Mvar-space and a Euclidean Template:Mvar-space are distinct objects, strictly speaking. Any Euclidean Template:Mvar-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 Template:Math, but it is not the only possible one. Actually, any positive-definite quadratic form Template:Mvar defines its own "distance" Template:Math, but it is not very different from the Euclidean one in the sense that <math display="block">\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}). </math>
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 Template:Math, or its affine transformation, does not magnify distances more than by some fixed Template:Math, and does not make distances smaller than Template:Math times, a fixed finite number times smaller.Template:Clarify
The aforementioned equivalence of metric functions remains valid if Template:Math is replaced with Template:Math, where Template:Mvar 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 Template:Math is not especially different from the Euclidean metric, Template:Math is not always distinguished from a Euclidean Template:Math-space even in professional mathematical works.
In algebraic and differential geometryEdit
Although the definition of a manifold does not require that its model space should be Template:Math, 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 manifold|differentiable Template:Mvar-dimensional manifold]] can be embedded into Template:Math.
Other appearancesEdit
Other structures considered on Template:Math include the one of a pseudo-Euclidean space, symplectic structure (even Template:Mvar), and contact structure (odd Template:Mvar). All these structures, although can be defined in a coordinate-free manner, admit standard (and reasonably simple) forms in coordinates.
Template:Math is also a real vector subspace of Template:Math which is invariant to complex conjugation; see also complexification.
Polytopes in RnEdit
Template:See also There are three families of polytopes which have simple representations in Template:Math spaces, for any Template:Mvar, and can be used to visualize any affine coordinate system in a real Template:Mvar-space. Vertices of a hypercube have coordinates Template:Math where each Template:Mvar 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 Template:Num and 1. An Template:Mvar-hypercube can be thought of as the Cartesian product of Template:Mvar identical intervals (such as the unit interval Template:Closed-closed) on the real line. As an Template:Mvar-dimensional subset it can be described with a [[system of inequalities|system of Template:Math inequalities]]: <math display="block">\begin{matrix} 0 \le x_1 \le 1 \\ \vdots \\ 0 \le x_n \le 1 \end{matrix}</math> for Template:Closed-closed, and <math display="block">\begin{matrix} |x_1| \le 1 \\ \vdots \\ |x_n| \le 1 \end{matrix}</math> for Template:Closed-closed.
Each vertex of the cross-polytope has, for some Template:Mvar, the Template:Mvar coordinate equal to ±1 and all other coordinates equal to 0 (such that it is the Template:Mvarth standard basis vector up to sign). This is a dual polytope of hypercube. As an Template:Mvar-dimensional subset it can be described with a single inequality which uses the absolute value operation: <math display="block">\sum_{k=1}^n |x_k| \le 1\,,</math> but this can be expressed with a system of Template:Math linear inequalities as well.
The third polytope with simply enumerable coordinates is the standard simplex, whose vertices are Template:Mvar standard basis vectors and the origin Template:Math. As an Template:Mvar-dimensional subset it is described with a system of Template:Math linear inequalities: <math display="block">\begin{matrix} 0 \le x_1 \\ \vdots \\ 0 \le x_n \\ \sum\limits_{k=1}^n x_k \le 1 \end{matrix}</math> Replacement of all "≤" with "<" gives interiors of these polytopes.
Topological propertiesEdit
The topological structure of Template:Math (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, Template:Math 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 Template:Math to itself which are not isometries, there can be many Euclidean structures on Template:Math 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 Template:Math onto itself, or its parts such as a Euclidean open ball or the interior of a hypercube).
Template:Math has the topological dimension Template:Mvar.
An important result on the topology of Template:Math, that is far from superficial, is Brouwer's invariance of domain. Any subset of Template:Math (with its subspace topology) that is homeomorphic to another open subset of Template:Math is itself open. An immediate consequence of this is that Template:Math is not homeomorphic to Template:Math if Template:Math – 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-dimensionalTemplate:Clarify real space continuously and surjectively onto Template:Math. A continuous (although not smooth) space-filling curve (an image of Template:Math) is possible.Template:Clarify
ExamplesEdit
| File:Real 0-space.svg |
| Empty column vector, the only element of Template:Math |
n ≤ 1Edit
Cases of Template:Math do not offer anything new: Template:Math is the real line, whereas Template:Math (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 cases of theories that describe different Template:Mvar.
n = 2Edit
Template:Details Template:See also The case of (x,y) where x and y are real numbers has been developed as the Cartesian plane P. Further structure has been attached with Euclidean vectors representing directed line segments in P. The plane has also been developed as the field extension <math>\mathbf{C}</math> by appending roots of X2 + 1 = 0 to the real field <math>\mathbf{R}.</math> The root i acts on P as a quarter turn with counterclockwise orientation. This root generates the group <math>\{i, -1, -i, +1\} \equiv \mathbf{Z}/4\mathbf{Z}</math>. When (x,y) is written x + y i it is a complex number.
Another group action by <math>\mathbf{Z}/2\mathbf{Z}</math>, where the actor has been expressed as j, uses the line y=x for the involution of flipping the plane (x,y) ↦ (y,x), an exchange of coordinates. In this case points of P are written x + y j and called split-complex numbers. These numbers, with the coordinate-wise addition and multiplication according to jj=+1, form a ring that is not a field.
Another ring structure on P uses a nilpotent e to write x + y e for (x,y). The action of e on P reduces the plane to a line: It can be decomposed into the projection into the x-coordinate, then quarter-turning the result to the y-axis: e (x + y e) = x e since e2 = 0. A number x + y e is a dual number. The dual numbers form a ring, but, since e has no multiplicative inverse, it does not generate a group so the action is not a group action.
Excluding (0,0) from P makes [x : y] projective coordinates which describe the real projective line, a one-dimensional space. Since the origin is excluded, at least one of the ratios x/y and y/x exists. Then [x : y] = [x/y : 1] or [x : y] = [1 : y/x]. The projective line P1(R) is a topological manifold covered by two coordinate charts, [z : 1] → z or [1 : z] → z, which form an atlas. For points covered by both charts the transition function is multiplicative inversion on an open neighborhood of the point, which provides a homeomorphism as required in a manifold. One application of the real projective line is found in Cayley–Klein metric geometry.
n = 3Edit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
n = 4Edit
Template:Details Template:Math can be imagined using the fact that Template:Num points Template:Math, where each Template:Mvar is either 0 or 1, are vertices of a tesseract (pictured), the 4-hypercube (see above).
The first major use of Template:Math 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 Template:Mvar 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 Template:Math with a curved metric for most practical purposes. None of these structures provide a (positive-definite) metric on Template:Math.
Euclidean Template:Math 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, Template:Math is the only case where Template:Math admits a non-standard differential structure: see exotic R4.
Norms on Template:MathEdit
One could define many norms on the vector space Template:Math. Some common examples are
- the p-norm, defined by <math display="inline">\|\mathbf{x}\|_p := \sqrt[p]{\sum_{i=1}^n|x_i|^p}</math> for all <math>\mathbf{x} \in \mathbf{R}^n</math> where <math>p</math> is a positive integer. The case <math>p = 2</math> is very important, because it is exactly the Euclidean norm.
- the <math>\infty</math>-norm or maximum norm, defined by <math>\|\mathbf{x}\|_\infty:=\max \{x_1,\dots,x_n\}</math> for all <math>\mathbf{x} \in \mathbf{R}^n</math>. This is the limit of all the p-norms: <math display="inline">\|\mathbf{x}\|_\infty = \lim_{p \to \infty} \sqrt[p]{\sum_{i=1}^n|x_i|^p}</math>.
A really surprising and helpful result is that every norm defined on Template:Math is equivalent. This means for two arbitrary norms <math>\|\cdot\|</math> and <math>\|\cdot\|'</math> on Template:Math you can always find positive real numbers <math>\alpha,\beta > 0</math>, such that <math display="block">\alpha \cdot \|\mathbf{x}\| \leq \|\mathbf{x}\|' \leq \beta\cdot\|\mathbf{x}\|</math> for all <math>\mathbf{x} \in \R^n</math>.
This defines an equivalence relation on the set of all norms on Template:Math. With this result you can check that a sequence of vectors in Template:Math converges with <math>\|\cdot\|</math> if and only if it converges with <math>\|\cdot\|'</math>.
Here is a sketch of what a proof of this result may look like:
Because of the equivalence relation it is enough to show that every norm on Template:Math is equivalent to the Euclidean norm <math>\|\cdot\|_2</math>. Let <math>\|\cdot\|</math> be an arbitrary norm on Template:Math. The proof is divided in two steps:
- We show that there exists a <math>\beta > 0</math>, such that <math>\|\mathbf{x}\| \leq \beta \cdot \|\mathbf{x}\|_2</math> for all <math>\mathbf{x} \in \mathbf{R}^n</math>. In this step you use the fact that every <math>\mathbf{x} = (x_1, \dots, x_n) \in \mathbf{R}^n</math> can be represented as a linear combination of the standard basis: <math display="inline">\mathbf{x} = \sum_{i=1}^n e_i \cdot x_i</math>. Then with the Cauchy–Schwarz inequality <math display="block">\|\mathbf{x}\| = \left\|\sum_{i=1}^n e_i \cdot x_i \right\|\leq \sum_{i=1}^n \|e_i\| \cdot |x_i|
\leq \sqrt{\sum_{i=1}^n \|e_i\|^2} \cdot \sqrt{\sum_{i=1}^n |x_i|^2} = \beta \cdot \|\mathbf{x}\|_2,</math> where <math display="inline">\beta := \sqrt{\sum_{i=1}^n \|e_i\|^2}</math>.
- Now we have to find an <math>\alpha > 0</math>, such that <math>\alpha\cdot\|\mathbf{x}\|_2 \leq \|\mathbf{x}\|</math> for all <math>\mathbf{x} \in \mathbf{R}^n</math>. Assume there is no such <math>\alpha</math>. Then there exists for every <math>k \in \mathbf{N}</math> a <math>\mathbf{x}_k \in \mathbf{R}^n</math>, such that <math>\|\mathbf{x}_k\|_2 > k \cdot \|\mathbf{x}_k\|</math>. Define a second sequence <math>(\tilde{\mathbf{x}}_k)_{k \in \mathbf{N}}</math> by <math display="inline">\tilde{\mathbf{x}}_k := \frac{\mathbf{x}_k}{\|\mathbf{x}_k\|_2}</math>. This sequence is bounded because <math>\|\tilde{\mathbf{x}}_k\|_2 = 1</math>. So because of the Bolzano–Weierstrass theorem there exists a convergent subsequence <math>(\tilde{\mathbf{x}}_{k_j})_{j\in\mathbf{N}}</math> with limit <math>\mathbf{a} \in</math> Template:Math. Now we show that <math>\|\mathbf{a}\|_2 = 1</math> but <math>\mathbf{a} = \mathbf{0}</math>, which is a contradiction. It is <math display="block">\|\mathbf{a}\| \leq \left\|\mathbf{a} - \tilde{\mathbf{x}}_{k_j}\right\| + \left\|\tilde{\mathbf{x}}_{k_j}\right\| \leq \beta \cdot \left\|\mathbf{a} - \tilde{\mathbf{x}}_{k_j}\right\|_2 + \frac{\|\mathbf{x}_{k_j}\|}{\|\mathbf{x}_{k_j}\|_2} \ \overset{j \to \infty}{\longrightarrow} \ 0,</math> because <math>\|\mathbf{a}-\tilde{\mathbf{x}}_{k_j}\| \to 0</math> and <math>0 \leq \frac{\|\mathbf{x}_{k_j}\|}{\|\mathbf{x}_{k_j}\|_2} < \frac{1}{k_j}</math>, so <math>\frac{\|\mathbf{x}_{k_j}\|}{\|\mathbf{x}_{k_j}\|_2} \to 0</math>. This implies <math>\|\mathbf{a}\| = 0</math>, so <math>\mathbf{a}= \mathbf{0}</math>. On the other hand <math>\|\mathbf{a}\|_2 = 1</math>, because <math>\|\mathbf{a}\|_2 = \left\| \lim_{j \to \infty}\tilde{\mathbf{x}}_{k_j} \right\|_2 = \lim_{j \to \infty} \left\| \tilde{\mathbf{x}}_{k_j} \right\|_2 = 1</math>. This can not ever be true, so the assumption was false and there exists such a <math>\alpha > 0</math>.
See alsoEdit
- Exponential object, for theoretical explanation of the superscript notation
- Geometric space
- Real projective space