In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector on one basis is, in general, different from the coordinate vector that represents on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis. Such a conversion results from the change-of-basis formula which expresses the coordinates relative to one basis in terms of coordinates relative to the other basis. Using matrices, this formula can be written where "old" and "new" refer respectively to the firstly defined basis and the other basis, and are the column vectors of the coordinates of the same vector on the two bases, and is the change-of-basis matrix (also called transition matrix), which is the matrix whose columns are the coordinate vectors of the new basis vectors on the old basis. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Change of basis formula Let be a basis of a finite-dimensional vector space over a field . For , one can define a vector by its coordinates over Let be the matrix whose th column is formed by the coordinates of . (Here and in what follows, the index refers always to the rows of and the while the index refers always to the columns of and the such a convention is useful for avoiding errors in explicit computations.) Setting one has that is a basis of if and only if the matrix is invertible, or equivalently if it has a nonzero determinant. In this case, is said to be the change-of-basis matrix from the basis to the basis Given a vector let be the coordinates of over and its coordinates over that is (One could take the same summation index for the two sums, but choosing systematically the indexes for the old basis and for the new one makes clearer the formulas that follows, and helps avoiding errors in proofs and explicit computations.) The change-of-basis formula expresses the coordinates over the old basis in term of the coordinates over the new basis. With above