on the left and on the right, one gets It follows that, for one has which is the change-of-basis formula expressed in terms of linear maps instead of coordinates. Function defined on a vector space A function that has a vector space as its domain is commonly specified as a multivariate function whose variables are the coordinates on some basis of the vector on which the function is applied. When the basis is changed, the expression of the function is changed. This change can be computed by substituting the "old" coordinates for their expressions in terms of the "new" coordinates. More precisely, if is the expression of the function in terms of the old coordinates, and if is the change-of-base formula, then is the expression of the same function in terms of the new coordinates. The fact that the change-of-basis formula expresses the old coordinates in terms of the new one may seem unnatural, but appears as useful, as no matrix inversion is needed here. As the change-of-basis formula involves only linear functions, many function properties are kept by a change of basis. This allows defining these properties as properties of functions of a variable vector that are not related to any specific basis. So, a function whose domain is a vector space or a subset of it is a linear function, a polynomial function, a continuous function, a differentiable function, a smooth function, an analytic function, if the multivariate function that represents it on some basis—and thus on every basis—has the same property. This is specially useful in the theory of manifolds, as this allows extending the concepts of continuous, differentiable, smooth and analytic functions to functions that are defined on a manifold. Linear maps Consider a linear map from a vector space of dimension to a vector space of dimension . It is represented on "old" bases of and by a matrix . A change of bases is defined by an change-of-basis matrix for , and an change-of-basis matrix for . On the "new" bases, the matrix of is This is a straightforward consequence of the change-of-basis formula. Endomorphisms Endomorphisms, are linear maps from a vector space to itself. For a change of basis, the formula of the preceding section applies, with the same change-of-basis matrix on both sides of the formula. That is, if is the square matrix of an endomorphism of over an "old" basis, and