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 Fn. Coordinate vectors allow calculations with abstract objects to be transformed into calculations with blocks of numbers (matrices and column vectors).
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.
In this case, transformations are obtained with a post-multiplication by a transformation matrix.
We can mechanize the above transformation by defining a function φB, called the standard representation of V with respect to B, that takes every vector to its coordinate representation: φB(v) = [v]B. Then φB is a linear transformation from V to Fn. 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 φB 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:
then the corresponding coordinate vector to the polynomial
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 b1, b2, ..., bn:
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:
The matrix M is an invertible matrix and M-1 is the basis transformation matrix from C to B. In other words,