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Linear Algebra - Linear Transformations, Change of Basis

  1. Sep 14, 2009 #1
    1. The problem statement, all variables and given/known data

    I need to prove this formula, but I'm not sure how to prove it.


    [T]C = P(C<-B).[T]B.P(C<-B)-1

    whereby B and C are bases in finite dimensional vector space V, and T is a linear transformation. Your help is greatly appreciated!

    2. Relevant equations

    T(x)=Ax
    [x]C=P(C<-B)[x]B
    Similar matrices.

    3. The attempt at a solution

    T(x)=Ax
    [T]C=[[T(c1)]C...[T(cn)]C

    ...
     
  2. jcsd
  3. Sep 14, 2009 #2

    HallsofIvy

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    Suppose [x]B is some vector x represented in basis B and that Tx= y (independent of the basis).

    Then, by definition of "PB<-C", PC<-B is P-1B<-C is so P^{-1}B<-C[x]C= [x]B.

    Then TBP^{-1}B<-C[x]CxC= TB[x]B= [y]B and, finally, PB<-CTBP^{-1}B<-C[x]C= PB<-C[y]C= [y]_B.
     
  4. Sep 15, 2009 #3


    Okay, thanks! I did manage to get something like that before, but I got stuck because I didn't do the tx=y thing.

    But may I ask, why does Tx= y need to be independent of the basis?
     
  5. Sep 15, 2009 #4

    HallsofIvy

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    A linear transformation, T, is a function from vector space U to vector space V such that T(au+bv)= aT(u)+ bT(v). There are no "bases" required for its definition. Bases are only necessary to write the linear transformation as a matrix.
     
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