notation, it is In terms of matrices, the change of basis formula is where and are the column vectors of the coordinates of over and respectively. Proof: Using the above definition of the change-of basis matrix, one has As the change-of-basis formula results from the uniqueness of the decomposition of a vector over a basis. Example Consider the Euclidean vector space Its standard basis consists of the vectors and If one rotates them by an angle of , one gets a new basis formed by and So, the change-of-basis matrix is The change-of-basis formula asserts that, if are the new coordinates of a vector then one has That is, This may be verified by writing In terms of linear maps Normally, a matrix represents a linear map, and the product of a matrix and a column vector represents the function application of the corresponding linear map to the vector whose coordinates form the column vector. The change-of-basis formula is a specific case of this general principle, although this is not immediately clear from its definition and proof. When one says that a matrix represents a linear map, one refers implicitly to bases of implied vector spaces, and to the fact that the choice of a basis induces an isomorphism between a vector space and , where is the field of scalars. When only one basis is considered for each vector space, it is worth to leave this isomorphism implicit, and to work up to an isomorphism. As several bases of the same vector space are considered here, a more accurate wording is required. Let be a field, the set of the -tuples is a -vector space whose addition and scalar multiplication are defined component-wise. Its standard basis is the basis that has as its th element the tuple with all components equal to except the th that is . A basis of a -vector space defines a linear isomorphism by Conversely, such a linear isomorphism defines a basis, which is the image by of the standard basis of Let be the "old basis" of a change of basis, and the associated isomorphism. Given a change-of basis matrix , let consider it as the matrix of an endomorphism of Finally, let define (where denotes function composition), and A straightforward verification, allows showing that this definition of is the same as that of the preceding section. Now, by composing the equation with