An array of numbers can be used to represent an element of a vector space.

From the beginning we have adopted the convention that vectors in are represented by matrices - column vectors - with entries in . Another, equivalent way to arrive at such a description is to start with the standard basis , viewed simply as a linearly independent set, and then define to be the span of these basis vectors. In this way, we realize that when writing down an column vector representing , we are simply recording (in columnar form) the coefficients that occur when representing as a linear combination of standard basis vectors: We notice now that this can be done not just for the standard basis in , but for any basis in any vector space. For the purposes of this section and what follows, we will only be concerned with such representations in finite dimensional vector spaces (or subspaces).

It is important here to distinguish between i) a vector and ii) its coordinate representation with respect to a particular basis for . For that reason, vectors in - written as column vectors - may some times be written with the decoration indicating that we are really looking at the representation of the vector with respect to the standard basis. As we will see, this additional decoration provides necessary book-keeping in the event we are considering representations with respect to different bases in

This last example illustrates one of the basic computations we will want to be able to do; given the coordinate description of a vector with respect to one basis, find its coordinate representation with respect to a (specified) different basis. The matrix in the above example is referred to as a transition matrix; specifically the base transition matrix from the basis to the basis , where As we will see in more detail below, the notation for the base transition matrix from to is

For now we will simply define base transition matrices. The proof of the equality in equation (eqn:transition) below will be given in the next section.

The discussion of base change in applies more generally to transitioning between two bases for an arbitrary vector space . The setup for this (and the proof that it works as claimed) is best handled within the more general framework of linear transformations and their matrix representations, which we discuss in the next section.