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In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis.^[1] An easy example may be a position such as (5, 2, 1) in a 3-dimensional Cartesian coordinate system with the basis as the axes of this system. Coordinates are always specified relative to an ordered basis. Bases and their associated coordinate representations let one realize vector spaces and linear transformations concretely as column vectors, row vectors, and matrices; hence, they are useful in calculations.

The idea of a coordinate vector can also be used for infinite-dimensional vector spaces, as addressed below.

Definition

Let V be a vector space of dimension n over a field F and let

${\displaystyle B=\{b_{1},b_{2},\ldots ,b_{n}\}}$

be an ordered basis for V. Then for every ${\displaystyle v\in V}$ there is a unique linear combination of the basis vectors that equals ${\displaystyle v}$:

${\displaystyle v=\alpha _{1}b_{1}+\alpha _{2}b_{2}+\cdots +\alpha _{n}b_{n}.}$

The coordinate vector of ${\displaystyle v}$ relative to B is the sequence of coordinates

${\displaystyle _{B}=(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}).}$

This is also called the representation of ${\displaystyle v}$ with respect to B, or the B representation of ${\displaystyle v}$. The ${\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}$ are called the coordinates of ${\displaystyle v}$. The order of the basis becomes important here, since it determines the order in which the coefficients are listed in the coordinate vector.

Coordinate vectors of finite-dimensional vector spaces can be represented by matrices as column or row vectors. In the above notation, one can write

${\displaystyle _{B}={\begin{bmatrix}\alpha _{1}\\\vdots \\\alpha _{n}\end{bmatrix}}}$

and

${\displaystyle _{B}^{T}={\begin{bmatrix}\alpha _{1}&\alpha _{2}&\cdots &\alpha _{n}\end{bmatrix}}}$

where ${\displaystyle _{B}^{T}}$ is the transpose of the matrix ${\displaystyle _{B}}$.

The standard representation

We can mechanize the above transformation by defining a function ${\displaystyle \phi _{B}}$, called the standard representation of V with respect to B, that takes every vector to its coordinate representation: ${\displaystyle \phi _{B}(v)=v_{B}}$. Then ${\displaystyle \phi _{B}}$ is a linear transformation from V to Fⁿ. In fact, it is an isomorphism, and its inverse ${\displaystyle \phi _{B}^{-1}:F^{n}\to V}$ is simply

${\displaystyle \phi _{B}^{-1}(\alpha _{1},\ldots ,\alpha _{n})=\alpha _{1}b_{1}+\cdots +\alpha _{n}b_{n}.}$

Alternatively, we could have defined ${\displaystyle \phi _{B}^{-1}}$ to be the above function from the beginning, realized that ${\displaystyle \phi _{B}^{-1}}$ is an isomorphism, and defined ${\displaystyle \phi _{B}}$ to be its inverse.

Examplesedit

Example 1edit

Let ${\displaystyle P_{3}}$ be the space of all the algebraic polynomials of degree at most 3 (i.e. the highest exponent of x can be 3). This space is linear and spanned by the following polynomials:

${\displaystyle B_{P}=\left\{1,x,x^{2},x^{3}\right\}}$

matching

${\displaystyle 1:=\begin{array}{c}1\\ 0\\ 0\\ 0\end{array};x:=\begin{array}{c}0\\ 1\\ 0\\ 0\end{array};x^2:=\begin{array}{c}0\\ 0\\ 1\\ 0\end{array};x^3:=\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}}$Zdroj:https://en.wikipedia.org?pojem=Coordinate_vector
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