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.
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.
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
relative to B is the sequence of coordinates This is also called the representation of
The order of the basis becomes important here, since it determines the order in which the coefficients are listed in the coordinate vector.
We can mechanize the above transformation by defining a function
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: matching then the coordinate vector corresponding 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 anti-Hermitian or neither, spectrum and eigenvalues, and more.
The Pauli matrices, which represent the spin operator when transforming the spin eigenstates into vector coordinates.
Let B and C be two different bases of a vector space V, and let us mark with
Any vector v represented in B can be transformed to a representation in C as follows: 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.
In other words, 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.