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A question about linear transformations

  1. Oct 11, 2012 #1
    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.
     
  2. jcsd
  3. Oct 12, 2012 #2

    micromass

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    Yes, the domain and range will remain the same.
     
  4. Oct 12, 2012 #3

    mathwonk

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    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.
     
    Last edited: Oct 12, 2012
  5. Oct 13, 2012 #4
    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]?
     
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