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Definition: Suppose we have two bases B=(u_1, u_2, ... , u_n) and B'=(v_1, v_2, ... , v_n) of the same vector space V, where B is the considered the old basis and B' the new basis. We note that every vector of the new basis B' is a linear combination of the old basis B. Then B'=BP, where the jth column of P is the coordinate vector of the new basis vector v_j with respect to the old basis. This matrix P is the basechange matrix.
Source: Algebra, second edition by Michael Artin