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 [T]_B : W \rightarrow W ?

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]

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.

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 [T]_B : V \rightarrow V?