 Coordinate vector

In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space F^{n}. Coordinate vectors allow calculations with abstract objects to be transformed into calculations with blocks of numbers (matrices, column vectors and row vectors).
The idea of a coordinate vector can also be used for infinite dimensional vector spaces, as addressed below.
Contents
Definition
Let V be a vector space of dimension n over a field F and let
be an ordered basis for V. Then for every there is a unique linear combination of the basis vectors that equals v:
The linear independence of vectors in the basis ensures that the αs are determined uniquely by v and B. Now, we define the coordinate vector of v relative to B to be the following sequence of coordinates:
This is also called the representation of v with respect of B, or the B representation of v. The αs are called the coordinates of 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 as elements of a column or row vector. This depends on the author's intention of preforming linear transformations by matrix multiplication on the left (premultiplication) or on the right (postmultiplication) of the vector. A column vector of length n can be premultiplied by any matrix with n columns, while a row vector of length n can be postmultiplied by any matrix with n rows.
For instance, a transformation from basis B to basis C may be obtained by premultiplying the column vector [v]_{B} by a square matrix (see below), resulting in a column vector :
If [v]_{B} is a row vector instead of a column vector, the same basis transformation can be obtained by postmultiplying the row vector by the transposed matrix to obtain the row vector :
The standard representation
We can mechanize the above transformation by defining a function ϕ_{B}, called the standard representation of V with respect to B, that takes every vector to its coordinate representation: ϕ_{B}(v) = [v]_{B}. Then ϕ_{B} is a linear transformation from V to F^{n}. In fact, it is an isomorphism, and its inverse is simply
Alternatively, we could have defined to be the above function from the beginning, realized that is an isomorphism, and defined ϕ_{B} to be its inverse.
Examples
Example 1
Let P4 be the space of all the algebraic polynomials in degree less than 4 (i.e. the highest exponent of x can be 3). This space is linear and spanned by the following polynomials:
 B_{P} = {1,x,x^{2},x^{3}}
matching
then the corresponding coordinate vector to the polynomial
 is .
According to that representation, the differentiation operator d/dx which we shall mark D will be represented by the following matrix:
Using that method it is easy to explore the properties of the operator: such as invertibility, hermitian or antihermitian or none, spectrum and eigenvalues and more.
Example 2
The Pauli matrices which represent the spin operator when transforming the spin eigenstates into vector coordinates.
Basis transformation matrix
Let B and C be two different bases of a vector space V, and let us mark with the matrix which has columns consisting of the C representation of basis vectors b_{1}, b_{2}, ..., b_{n}:
This matrix is referred to as the basis transformation matrix from B to C, and can be used for transforming any vector v from a B representation to a C representation, according to the following theorem:
If E is the standard basis, the transformation from B to E can be represented with the following simplified notation:
where
 and
Corollary
The matrix M is an invertible matrix and M^{1} is the basis transformation matrix from C to B. In other words,
Remarks
 The basis transformation matrix can be regarded as an automorphism over V.
 In order to easily remember the theorem

 notice that M 's superscript and v 's subscript indices are "canceling" each other and M 's subscript becomes v 's new subscript. This "canceling" of indices is not a real canceling but rather a convenient and intuitively appealing, although mathematically incorrect, manipulation of symbols, permitted by an appropriately chosen notation.
Infinite dimensional vector spaces
Suppose V is an infinite dimensional vector space over a field F. If the dimension is κ, then there is some basis of κ elements for V. After an order is chosen, the basis can be considered an ordered basis. The elements of V are finite linear combinations of elements in the basis, which give rise to unique coordinate representations exactly as described before. The only change is that the indexing set for the coordinates is not finite. Since a given vector v is a finite linear combination of basis elements, the only nonzero entries of the coordinate vector for v will be the nonzero coefficients of the linear combination representing v. Thus the coordinate vector for v is zero except in finitely many entries.
The linear transformations between (possibly) infinite dimensional vector spaces can be modeled, analogously to the finite dimensional case, with infinite matrices. The special case of the transformations from V into V is described in the full linear ring article.
Categories: Linear algebra
 Vectors
Wikimedia Foundation. 2010.
Look at other dictionaries:
Vector — may refer to: In mathematics * Euclidean vector, a geometric entity endowed with both length and direction, an element of a Euclidean vector space * Coordinate vector, in linear algebra, an explicit representation of an element of any abstract… … Wikipedia
Vector space — This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry). Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is… … Wikipedia
Vector notation — This page is an overview of the common notations used when working with vectors, which may be spatial or more abstract members of vector spaces.The common typographic convention for representing a vector is upright boldface type, as in v for a… … Wikipedia
Vector control (motor) — Vector control (also called Field Oriented Control, FOC) is one method used in variable frequency drives to control the torque (and thus finally the speed) of three phase AC electric motors by controlling the current fed to the machine.MethodThe… … Wikipedia
Vector decomposition — refers to decomposing a vector of Rn into several vectors, each linearly independent (in mutually distinct directions in the n dimensional space). Vector decomposition in two dimensions In two dimensions, a vector can be decomposed in many ways.… … Wikipedia
Coordinate — Co*[ o]r di*nate, n. 1. A thing of the same rank with another thing; one two or more persons or things of equal rank, authority, or importance. [1913 Webster] It has neither co[ o]rdinate nor analogon; it is absolutely one. Coleridge. [1913… … The Collaborative International Dictionary of English
Vector field reconstruction — [ [http://prola.aps.org/pdf/PRE/v51/i5/p4262 1 Global Vector Field Reconstruction from a Chaotic Experimental Signal in Copper Electrodissolution.] Letellier C, Le Sceller L , Maréchal E, Dutertre P, Maheu B, Gouesbet G, Fei Z, Hudson JL.… … Wikipedia
vector — [vek′tər] n. [ModL < L, bearer, carrier < vectus, pp. of vehere, to carry: see WAY] 1. Biol. an animal, esp. an insect, that transmits a disease producing organism from a host to a noninfected animal 2. Math. a) a mathematical expression… … English World dictionary
Vector field — In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid… … Wikipedia
Coordinate space — In mathematics, specifically in linear algebra, the coordinate space, Fn, is the prototypical example of an n dimensional vector space over a field F. It can be defined as the product space of F over a finite index set. Contents 1 Definition 1.1… … Wikipedia