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]

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 [itex][T]_B : V \rightarrow V[/itex]?

 

[blue_raver22]

Yes, the domain and range will still be W if we write T with respect to a different basis B. This is because a linear transformation is defined by its effect on the vector space, not by the basis we use to represent it. The notation [T]_B : W \rightarrow W simply means that we are representing the linear transformation T with respect to the basis B, but it does not change the underlying vector space W.

To understand this concept further, consider the analogy of a map. A map represents the same geographical area regardless of the language used to label the places on the map. Similarly, a linear transformation represents the same vector space regardless of the basis used to represent it.

In mathematics, we often use different bases to represent the same vector space because it can make certain calculations or proofs easier. However, the underlying vector space remains the same. So, the notation [T]_B : W \rightarrow W simply indicates that we are representing the linear transformation T with respect to the basis B, but it does not change the domain and range of the transformation.

I hope this explanation helps to clarify any confusion.