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. 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 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 : The coordinate vector of relative to B is the sequence of coordinates This is also called the representation of with respect to B, or the B representation of . The are called the coordinates of . 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 and where is the transpose of the matrix . The standard representation We can mechanize the above transformation by defining a function , called the standard representation of V with respect to B, that takes every vector to its coordinate representation: . Then 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 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: 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: