Coordinate space

The coordinate space in two real dimensions consists of all vectors that have the coordinate origin as the starting point

The coordinate space , standard space or standard vector space in mathematics is the vector space of - tuples with components from a given field provided with the component-wise addition and scalar multiplication . The elements of the coordinate space are called coordinate vectors or coordinate tuples. The standard basis for the coordinate space consists of the canonical unit vectors . Linear mappings between coordinate spaces are represented by matrices . The coordinate spaces have a special meaning in linear algebra , since every finite-dimensional vector space is isomorphic (structurally identical) to a coordinate space . ${\displaystyle n}$

The two- and three-dimensional real coordinate spaces often serve as models for the Euclidean plane and the three-dimensional Euclidean space . In this case, its elements are understood both as points and as vectors .

definition

A coordinate vector (x, y, z) as a position vector in three-dimensional real coordinate space

Is a field and a natural number , then is -fold the Cartesian product ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle K^{n}=\{(x_{1},\ldots ,x_{n})\mid x_{1},\ldots ,x_{n}\in K\}}$

the set of all - tuples with components . For this tuple is now defined a component-wise addition of ${\displaystyle n}$ ${\displaystyle (x_{1},\ldots ,x_{n})}$ ${\displaystyle K}$ ${\displaystyle +\colon K^{n}\times K^{n}\to K^{n}}$

${\displaystyle (x_{1},\ldots ,x_{n})+(y_{1},\ldots ,y_{n})=(x_{1}+y_{1},\ldots ,x_{n}+y_{n})}$

as well as a component-wise multiplication with a scalar by ${\displaystyle \cdot \colon K\times K^{n}\to K^{n}}$

${\displaystyle a\cdot (x_{1},\ldots ,x_{n})=(a\cdot x_{1},\ldots ,a\cdot x_{n})}$.

In this way a vector space is obtained , which is called the coordinate space or the standard space of the dimension above the body . Its elements are called coordinate vectors or coordinate tuples. ${\displaystyle (K^{n},+,\cdot )}$ ${\displaystyle n}$ ${\displaystyle K}$

Representation with column vectors

The coordinate vectors are often noted as column vectors . The vector addition and scalar multiplication then correspond to a line-by-line addition of the vector components or a line-by-line multiplication with a scalar:

${\displaystyle {\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}+{\begin{pmatrix}y_{1}\\\vdots \\y_{n}\end{pmatrix}}={\begin{pmatrix}x_{1}+y_{1}\\\vdots \\x_{n}+y_{n}\end{pmatrix}},\quad a\cdot {\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}a\cdot x_{1}\\\vdots \\a\cdot x_{n}\end{pmatrix}}}$.

These operations are then special cases of matrix addition and scalar multiplication of single-column matrices .

Examples

Addition of two vectors in the Euclidean plane (above) and multiplication of a vector by the number two (below).

Important examples of coordinate spaces arise from the choice of real numbers as the underlying body. In one-dimensional coordinate space , the vector space operations correspond precisely to the normal addition and multiplication of numbers. In the two-dimensional real coordinate space , number pairs can be interpreted as position vectors in the Euclidean plane . The two components are then precisely the coordinates of the end point of a position vector in a Cartesian coordinate system . In this way the vector addition corresponds ${\displaystyle \mathbb {R} ^{1}}$ ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle {\vec {v}}+{\vec {w}}={\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}+{\begin{pmatrix}w_{1}\\w_{2}\end{pmatrix}}={\begin{pmatrix}v_{1}+w_{1}\\v_{2}+w_{2}\end{pmatrix}}}$

illustrative of the addition of the associated vector arrows and the multiplication of a vector by a number

${\displaystyle a\cdot {\vec {v}}=a\cdot {\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}={\begin{pmatrix}a\cdot v_{1}\\a\cdot v_{2}\end{pmatrix}}}$.

the stretching (or compression) of the associated vector arrow by the factor . In particular, a vector in the Euclidean plane is obtained again by vector addition or scalar multiplication. Correspondingly, the tuples of the three-dimensional real coordinate space can be interpreted as position vectors in Euclidean space . In higher dimensions, this construction works in a completely analogous manner, even if the coordinate vectors of the can then no longer be interpreted so clearly. ${\displaystyle a}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{n}}$

properties

Neutral and inverse element

The neutral element in the coordinate space is the zero vector

${\displaystyle (0,\ldots ,0)}$,

where is the zero element of the body . The element inverse to a vector is then the vector ${\displaystyle 0}$ ${\displaystyle K}$ ${\displaystyle (x_{1},\ldots ,x_{n})}$

${\displaystyle (-x_{1},\ldots ,-x_{n})}$,

wherein for the additive inverse element to each in is. ${\displaystyle -x_{i}}$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle x_{i}}$ ${\displaystyle K}$

Laws

The coordinate space satisfies the axioms of a vector space . In addition to the existence of a neutral and inverse element, coordinate vectors and scalars apply ${\displaystyle x,y,z\in K^{n}}$ ${\displaystyle a,b\in K}$

• the associative law , ${\displaystyle x+(y+z)=(x+y)+z}$
• the commutative law , ${\displaystyle x+y=y+x}$
• the mixed associative law , ${\displaystyle a\cdot (b\cdot x)=(a\cdot b)\cdot x}$
• the distributive laws and as well ${\displaystyle a\cdot (x+y)=a\cdot x+a\cdot y}$ ${\displaystyle (a+b)\cdot x=a\cdot x+b\cdot x}$
• the neutrality of one , being the one element of the body . ${\displaystyle 1\cdot x=x}$ ${\displaystyle 1}$ ${\displaystyle K}$

These laws follow directly from the associativity, commutativity and distributivity of addition and multiplication in the body by applying to each component of a coordinate tuple. ${\displaystyle K}$

Base

The standard basis for the coordinate space consists of the canonical unit vectors

${\displaystyle \{e_{1},e_{2},\dotsc ,e_{n}\}=\{(1,0,\dotsc ,0),(0,1,0,\dotsc ,0),\dotsc ,(0,\dotsc ,0,1)\}}$.

Each vector can thus be used as a linear combination ${\displaystyle x\in K^{n}}$

${\displaystyle x=x_{1}\cdot e_{1}+\dotsb +x_{n}\cdot e_{n}}$

represent the basis vectors. The dimension of the coordinate space is therefore given by

${\displaystyle \dim(K^{n})=n}$.

Further bases of the coordinate space can be determined by basic transformation of the standard basis. The column or row vectors of a matrix form a basis of the coordinate space exactly if the matrix is regular , i.e. has full rank . ${\displaystyle (n\times n)}$ ${\displaystyle K^{n}}$

Linear maps

The linear mappings between two coordinate spaces clearly correspond to the matrices with entries from the body: If a matrix has rows and columns, then the matrix-vector product becomes a linear map ${\displaystyle A\in K^{m\times n}}$ ${\displaystyle m}$ ${\displaystyle n}$

${\displaystyle f_{A}\colon K^{n}\to K^{m},\quad f_{A}(x)=A\cdot x}$

Are defined. Conversely, there is a clearly defined mapping matrix for every linear mapping , so that for all . The columns of result as the images of the standard basis vectors: ${\displaystyle f\colon K^{n}\to K^{m}}$ ${\displaystyle A_{f}\in K^{m\times n}}$ ${\displaystyle f(x)=A_{f}\cdot x}$ ${\displaystyle x\in K^{n}}$ ${\displaystyle A_{f}}$

${\displaystyle A_{f}={\bigl (}f(e_{1})\mid \cdots \mid f(e_{n}){\bigr )}}$.

The set of matrices together with the matrix addition and the scalar multiplication itself form a vector space, the matrix space .

Isomorphism

If there is an arbitrary -dimensional vector space over the body , then is isomorphic to the corresponding coordinate space , i.e. ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle K^{n}}$

${\displaystyle V\cong K^{n}}$.

If one chooses a basis for , then every vector has the representation ${\displaystyle \{b_{1},\dotsc ,b_{n}\}}$ ${\displaystyle V}$ ${\displaystyle v\in V}$

${\displaystyle v=c_{1}\cdot b_{1}+\dotsb +c_{n}\cdot b_{n}}$

with . Each vector can be clearly represented as a coordinate tuple. Conversely, because of the linear independence of the base vectors, exactly one vector corresponds to each such coordinate tuple . So the figure is ${\displaystyle c_{1},\dotsc ,c_{n}\in K}$ ${\displaystyle v\in V}$ ${\displaystyle (c_{1},\dotsc ,c_{n})\in K^{n}}$ ${\displaystyle V}$

${\displaystyle K^{n}\to V,\quad (c_{1},\dotsc ,c_{n})\mapsto c_{1}\cdot b_{1}+\dotsb +c_{n}\cdot b_{n}}$

bijective . Since the mapping is also linear, it represents an isomorphism between the coordinate space and the vector space . Since every -dimensional vector space over the body is isomorphic to the coordinate space in this way , all -dimensional vector spaces over the same body are isomorphic with one another. ${\displaystyle K^{n}}$ ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle K^{n}}$ ${\displaystyle n}$

This identification of finite-dimensional vector spaces with the associated coordinate space also explains the name “standard space”. Nevertheless, in linear algebra one often prefers to work with abstract vector spaces instead of coordinate spaces, since in theory one would like to argue without coordinates, i.e. without a specially selected basis. For concrete calculations, one then falls back on the coordinate space and calculates with the coordinate vectors.

Extensions

The coordinate space can be expanded to include the following mathematical structures , for example:

• Is a real or complex coordinate space with a scalar product is provided, for example, the standard scalar , one obtains a scalar product . Since this space with respect to the induced by the scalar metric completely is, these are even a Hilbert space .

• If a real or complex coordinate space is given a vector norm , for example the Euclidean norm or another p norm , a normalized space is obtained . This space is then also complete with regard to the metric induced by the norm , i.e. a Banach space .

• If a coordinate space is provided with a topology , for example the standard topology , a topological vector space is obtained , that is, the vector addition and the scalar multiplication are continuous operations.

literature

• Gerd Fischer : Linear Algebra: an introduction for first-year students . Springer, 2008, ISBN 3-8348-9574-1 .
• Herbert Amann, Joachim Escher: Analysis I . Springer, 2006, ISBN 3-7643-7756-9 .

Individual evidence

1. ↑ Fischer: Linear Algebra: An Introduction for New Students . S. 75 .
2. ↑ ^{a b} Amann, Escher: Analysis I . S. 125 .