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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, a coordinate vector is a representation of a vector as an ordered list of numbers (a
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
) that describes the vector in terms of a particular ordered basis. 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 In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
''n'' over a field ''F'' and let : B = \ be an ordered basis for ''V''. Then for every v \in V there is a unique linear combination of the basis vectors that equals '' v '': : v = \alpha _1 b_1 + \alpha _2 b_2 + \cdots + \alpha _n b_n . The coordinate vector of '' v '' relative to ''B'' is the
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
: B = (\alpha _1, \alpha _2, \ldots, \alpha _n) . This is also called the ''representation of v with respect to B'', or the ''B representation of v ''. The \alpha _1, \alpha _2, \ldots, \alpha _n 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 matrices as column or row vectors. In the above notation, one can write : B = \begin \alpha_1 \\ \vdots \\ \alpha_n \end and : B^T = \begin \alpha_1 & \alpha_2 & \cdots & \alpha_n \end where B^T is the transpose of the matrix B.


The standard representation

We can mechanize the above transformation by defining a function \phi_B, called the ''standard representation of V with respect to B'', that takes every vector to its coordinate representation: \phi_B(v)= B. Then \phi_B is a linear transformation from ''V'' to ''F''''n''. In fact, it is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, and its
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
\phi_B^:F^n\to V is simply :\phi_B^(\alpha_1,\ldots,\alpha_n)=\alpha_1 b_1+\cdots+\alpha_n b_n. Alternatively, we could have defined \phi_B^ to be the above function from the beginning, realized that \phi_B^ is an isomorphism, and defined \phi_B to be its inverse.


Examples


Example 1

Let P3 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: :B_P = \left\ matching : 1 := \begin 1 \\ 0 \\ 0 \\ 0 \end ; \quad x := \begin 0 \\ 1 \\ 0 \\ 0 \end ; \quad x^2 := \begin 0 \\ 0 \\ 1 \\ 0 \end ; \quad x^3 := \begin 0 \\ 0 \\ 0 \\ 1 \end 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 is :\begin a_0 \\ a_1 \\ a_2 \\ a_3 \end. According to that representation, the differentiation operator ''d''/''dx'' which we shall mark ''D'' will be represented by the following
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
: :Dp(x) = P'(x) ; \quad = \begin 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ \end Using that method it is easy to explore the properties of the operator, such as: invertibility, Hermitian or anti-Hermitian or neither, 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 \lbrack M \rbrack_C^B the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
which has columns consisting of the ''C'' representation of basis vectors ''b1, b2, …, bn'': :\lbrack M\rbrack_C^B = \begin \lbrack b_1\rbrack_C & \cdots & \lbrack b_n\rbrack_C \end This matrix is referred to as the basis transformation matrix from ''B'' to ''C''. It can be regarded as an automorphism over F^n. Any vector ''v'' represented in ''B'' can be transformed to a representation in ''C'' 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

The matrix ''M'' is an invertible matrix and ''M''−1 is the basis transformation matrix from ''C'' to ''B''. In other words, :\begin \operatorname &= \lbrack M\rbrack_C^B \lbrack M\rbrack_B^C = \lbrack M\rbrack_C^C \\ pt &= \lbrack M\rbrack_B^C \lbrack M\rbrack_C^B = \lbrack M\rbrack_B^B \end


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.


See also

* Change of basis


References

{{reflist Linear algebra Vectors (mathematics and physics)