Coordinate Vector - Definition

Definition

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:

The linear independence of vectors in the basis ensures that the α-s 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 α-s 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 as elements of a column or row vector. This depends on the author's intention of performing linear transformations by matrix multiplication on the left (pre-multiplication) or on the right (post-multiplication) of the vector. A column vector of length n can be pre-multiplied by any matrix with n columns, while a row vector of length n can be post-multiplied by any matrix with n rows.

For instance, a transformation from basis B to basis C may be obtained by pre-multiplying the column vector by a square matrix (see below), resulting in a column vector :

If is a row vector instead of a column vector, the same basis transformation can be obtained by post-multiplying the row vector by the transposed matrix to obtain the row vector :