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Coordinate vector

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Concept in linear algebra

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Inlinear algebra, acoordinate vector is a representation of avector as an ordered list of numbers (atuple) that describes the vector in terms of a particularordered basis.^[1] An easy example may be a position such as (5, 2, 1) in a 3-dimensionalCartesian 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 realizevector spaces andlinear transformations concretely ascolumn vectors,row vectors, andmatrices; 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

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LetV be avector space ofdimensionn over afieldF and let

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

be anordered basis forV. Then for every $v\in V$ there is a uniquelinear combination of the basis vectors that equals $v {\displaystyle v}$ :

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

Thecoordinate vector of $v {\displaystyle v}$ relative toB is thesequence ofcoordinates

[v]_{B}=(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}).

This is also called therepresentation of $v {\displaystyle v}$ with respect to B, or theB representation of $v {\displaystyle v}$ . The $\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}$ are called thecoordinates of $v {\displaystyle v}$ . 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 bymatrices ascolumn orrow vectors. In the above notation, one can write

[v]_{B}={\begin{bmatrix}\alpha _{1}\\\vdots \\\alpha _{n}\end{bmatrix}}

and

[v]_{B}^{T}={\begin{bmatrix}\alpha _{1}&\alpha _{2}&\cdots &\alpha _{n}\end{bmatrix}}

where $[v]_{B}^{T}$ is thetranspose of the matrix $[v]_{B}$ .

The standard representation

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We can mechanize the above transformation by defining a function $\phi _{B}$ , called thestandard representation of V with respect to B, that takes every vector to its coordinate representation: $\phi _{B}(v)=[v]_{B}$ . Then $\phi _{B}$ is a linear transformation fromV toFⁿ. In fact, it is anisomorphism, and itsinverse $\phi _{B}^{-1}:F^{n}\to V$ is simply

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

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

Examples

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Example 1

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

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

matching

1:={\begin{bmatrix}1\\0\\0\\0\end{bmatrix}};\quad x:={\begin{bmatrix}0\\1\\0\\0\end{bmatrix}};\quad x^{2}:={\begin{bmatrix}0\\0\\1\\0\end{bmatrix}};\quad x^{3}:={\begin{bmatrix}0\\0\\0\\1\end{bmatrix}}

then the coordinate vector corresponding to the polynomial

p\left(x\right)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}

{\begin{bmatrix}a_{0}\\a_{1}\\a_{2}\\a_{3}\end{bmatrix}}.

According to that representation, thedifferentiation operatord/dx which we shall markD will be represented by the followingmatrix:

Dp(x)=P'(x);\quad [D]={\begin{bmatrix}0&1&0&0\\0&0&2&0\\0&0&0&3\\0&0&0&0\\\end{bmatrix}}

Using that method it is easy to explore the properties of the operator, such as:invertibility,Hermitian or anti-Hermitian or neither, spectrum andeigenvalues, and more.

Example 2

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ThePauli matrices, which represent thespin operator when transforming the spineigenstates into vector coordinates.

Basis transformation matrix

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LetB andC be two different bases of a vector spaceV, and let us mark with $\lbrack M\rbrack _{C}^{B}$ thematrix which has columns consisting of theC representation of basis vectorsb₁, b₂, …, b_n:

\lbrack M\rbrack _{C}^{B}={\begin{bmatrix}\lbrack b_{1}\rbrack _{C}&\cdots &\lbrack b_{n}\rbrack _{C}\end{bmatrix}}

This matrix is referred to as thebasis transformation matrix fromB toC. It can be regarded as anautomorphism over $F^{n}$ . Any vectorv represented inB can be transformed to a representation inC as follows:

\lbrack v\rbrack _{C}=\lbrack M\rbrack _{C}^{B}\lbrack v\rbrack _{B}.

Under the transformation of basis, notice that the superscript on the transformation matrix,M, and the subscript on the coordinate vector,v, are the same, and seemingly cancel, leaving the remaining subscript. While this may serve as a memory aid, it is important to note that no such cancellation, or similar mathematical operation, is taking place.

Corollary

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The matrixM is aninvertible matrix andM⁻¹ is the basis transformation matrix fromC toB. In other words,

{\begin{aligned}\operatorname {Id} &=\lbrack M\rbrack _{C}^{B}\lbrack M\rbrack _{B}^{C}=\lbrack M\rbrack _{C}^{C}\\[3pt]&=\lbrack M\rbrack _{B}^{C}\lbrack M\rbrack _{C}^{B}=\lbrack M\rbrack _{B}^{B}\end{aligned}}

Infinite-dimensional vector spaces

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SupposeV is an infinite-dimensional vector space over a fieldF. If the dimension isκ, then there is some basis ofκ elements forV. After an order is chosen, the basis can be considered an ordered basis. The elements ofV 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 vectorv is afinite linear combination of basis elements, the only nonzero entries of the coordinate vector forv will be the nonzero coefficients of the linear combination representingv. Thus the coordinate vector forv 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, withinfinite matrices. The special case of the transformations fromV intoV is described in thefull linear ring article.