The discussion revolves around the properties of linear transformations when expressed in different bases, specifically whether the domain and range of a linear transformation T:W -> W remain the same when represented as [T]_B with respect to a different basis B. The scope includes theoretical considerations of linear algebra and the implications of changing bases on the representation of transformations.
Participants express differing views on whether the domain and range of [T]_B can be considered as V or if they should be viewed as R^n, indicating that multiple competing perspectives remain unresolved.
The discussion highlights the dependence on definitions of basis and transformations, as well as the implications of viewing transformations in different coordinate systems. There are unresolved nuances regarding the interpretation of domain and range in the context of linear transformations.
mathwonk said:micromass is of course quite right in an intrinsic sense. nothing auxiliary changes the domain and range of a map.
still i look at these things a little differently. to me a basis is an isomorphism from the given vector space V to a coordinate space R^n.
then the matrix associated to that basis is a map from coordinate space to itself.
i.e. if T:V-->V, and the basis B defines the isom B:V-->R^n,
then we have [T]B:R^n-->R^n, where the composition
B^-1o[T]BoB: V-->R^n-->R^n-->V equals T:V-->V.
Thus in this sense, the domain of [T]B is R^n not V, and the range of [T]B is the image of the range of T under the isomorphism B:V-->R^n.