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 and column vectors).
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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:
By one of the defining properties of bases, 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.
Typically, but not necessarily, the coordinates are represented as elements of a column vector, so that they can be easily manipulated using matrix multiplication:
For instance, vector or basis transformations are obtained with a premultiplication of the column vector by a transformation matrix (see below). Some authors prefer using row vectors:
In this case, transformations are obtained with a postmultiplication by a transformation matrix.
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.
Let P3 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:
matching
then the corresponding coordinate vector to the polynomial
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.
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's 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
The matrix M is an invertible matrix and M^{1} is the basis transformation matrix from C to B. In other words,
