Editing Change of basis (section)

==Function defined on a vector space==

A [[function (mathematics)|function]] that has a vector space as its [[domain of a function|domain]] is commonly specified as a [[multivariate function]] whose variables are the coordinates on some basis of the vector on which the function is [[function application|applied]].

When the basis is changed, the [[expression (mathematics)|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 {{math|''f''('''x''')}} is the expression of the function in terms of the old coordinates, and if {{math|'''x''' {{=}} ''A'''''y'''}} is the change-of-base formula, then {{math|''f''(''A'''''y''')}} 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 function]]s, 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 [[manifold]]s, as this allows extending the concepts of continuous, differentiable, smooth and analytic functions to functions that are defined on a manifold.