A question about linear transformations...

Artusartos

If we have a linear transformation T:W -> W. Then if we write T with respect to a different basis B, will the domain and range still be W? So, will we have [itex][T]_B : W \rightarrow W[/itex] ?

If not, can anybody explain to me why?

Thanks in advance.

micromass

Yes, the domain and range will remain the same.

mathwonk

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.

Artusartos

I'm not sure if I understand this...You say that micromass's answer would be considered right, but then you say "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." So would it be wrong to say [itex][T]_B : V \rightarrow V[/itex]?