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In [[linear algebra]], the '''coordinate representation''' or '''coordinate vector''' of a vector is the unique tuple of numbers that describes the vector in terms of a particular [[ordered basis]]. Coordinates are always specified relative to an ordered basis. Bases and their associated coordinate representations let one realize [[vector space]]s and [[linear transformation]]s concretely as [[column vector]]s, [[row vector]]s, and [[matrix (mathematics)|matrices]], hence are useful in calculations.


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


== Definition ==
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Let ''V'' be a [[vector space]] of [[dimension (vector space)|dimension]] ''n'' over a [[field (mathematics)|field]] ''F'' and let
:<math> B = \{ b_1, b_2, \ldots, b_n \} </math>
be an [[ordered basis]] for ''V''.
Then for every <math> v \in V </math> there is a unique [[linear combination]] of the basis vectors that equals ''v'':
:<math> v = \alpha _1 b_1 + \alpha _2 b_2 + \cdots + \alpha _n b_n .</math>
The '''coordinate vector''' of ''v'' relative to ''B'' is the [[sequence]] of [[coordinates]]
:<math> [v]_B = (\alpha _1, \alpha _2, \cdots, \alpha _n) .</math>
This is also called the ''representation of v with respect of B'', or the ''B representation of v''. The α-s 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 by [[matrix_(mathematics)|matrices]] as [[column vector|column]]s or [[row vector]]s. In the above notation, one can write
:<math> [v]_B = \begin{bmatrix} \alpha_1 \\ \vdots \\ \alpha_n \end{bmatrix}</math>
or
:<math>[v]_B = [ \alpha_1, \dots, \alpha_n ].</math>
 
 
== The standard representation ==
We can mechanize the above transformation by defining a function <math>\phi_B</math>, called the ''standard representation of V with respect to B'', that takes every vector to its coordinate representation: <math>\phi_B(v)=[v]_B</math>. Then <math>\phi_B</math> is a linear transformation from ''V'' to ''F''<sup>n</sup>. In fact, it is an [[isomorphism]], and its [[inverse function|inverse]] <math>\phi_B^{-1}:F^n\to V</math> is simply
:<math>\phi_B^{-1}(\alpha_1,\ldots,\alpha_n)=\alpha_1 b_1+\cdots+\alpha_n b_n.</math>
 
Alternatively, we could have defined <math>\phi_B^{-1}</math> to be the above function from the beginning, realized that <math>\phi_B^{-1}</math> is an isomorphism, and defined <math>\phi_B</math> 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:
:<math> B_P = \{  1,  x,  x^2,  x^3 \} </math>
matching
:<math> 1 := \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \quad ; \quad x := \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \quad ; \quad x^2 := \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \quad ; \quad x^3 := \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \quad </math>
then the corresponding coordinate vector to the polynomial
:<math> p \left( x \right) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 </math>  is   <math> \begin{bmatrix} a_0 \\ a_1 \\ a_2 \\ a_3 \end{bmatrix} </math> .
According to that representation, the [[differentiation operator]] d/dx which we shall mark D will be represented by the following [[matrix (math)|matrix]]:
:<math> Dp(x) = P'(x) \quad ; \quad [D] =
\begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 0 & 0 & 3 \\
0 & 0 & 0 & 0 \\
\end{bmatrix}
</math>
Using that method it is easy to explore the properties of the operator: such as [[Invertible matrix|invertibility]], [[hermitian|hermitian or anti-hermitian or none]], spectrum and [[eigenvalues]] and more.
 
=== Example 2 ===
The [[Pauli matrices]] which represent the [[Spin (physics)|spin]] operator when transforming the spin [[eigenstate]]s into vector coordinates.
 
== Basis transformation matrix ==
Let ''B'' and ''C'' be two different bases of a vector space ''V'', and let us mark with <math> [M]_{C}^{B} </math> the [[Matrix (mathematics)|matrix]] which has columns consisting of the ''C'' representation of basis vectors ''b<sub>1</sub>,  b<sub>2</sub>, ..., b<sub>n</sub>'':
:<math> [M]_{C}^{B} =
\begin{bmatrix} \ [b_1]_C & \cdots & [b_n]_C \ \end{bmatrix}
</math>
 
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]]:
:<math> [v]_C = [M]_{C}^{B} [v]_B. </math>
 
If ''E'' is the [[standard basis]], the transformation from ''B'' to ''E'' can be represented with the following simplified notation:
 
:<math> v = [M]^B [v]_B. \,</math>
 
where
:<math> v = [v]_E, \,</math> and
 
:<math> [M]^B = [M]_{E}^B. </math>
 
===Corollary===
The matrix ''M'' is an [[invertible matrix]] and ''M<sup>-1</sup>'' is the basis transformation matrix from ''C'' to ''B''. In other words,
 
:<math> [M]_{C}^{B} [M]_{B}^{C} = [M]_{C}^{C} = \mathrm{Id} </math>
 
:<math> [M]_{B}^{C} [M]_{C}^{B} = [M]_{B}^{B} = \mathrm{Id} </math>
 
===Remarks===
# The basis transformation matrix can be regarded as an [[automorphism]] over ''V''.
# In order to easily remember the theorem
:: <math> [v]_C = [M]_{C}^{B} [v]_B,</math>  
::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_matrix#Infinite_matrices|infinite matrices]]. The special case of the transformations from ''V'' into ''V'' is described in the [[full linear ring]] article.
 
==See also==
* [[Change of basis]]
 
[[Category:Linear algebra]]
[[Category:Vectors]]

Revision as of 09:28, 22 February 2014


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