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Change of basis

  1. Oct 15, 2009 #1
    Hi everyone,

    I am working on the following problem.

    Suppose the set of vectors X1,..,Xk is a basis for linear space V1.
    Suppose the set of vectors Y1,..,Yk is also a basis for linear space
    V1.
    Clearly the linear space spanned by the Xs equals the linear space
    spanned by the Ys.

    Set
    X=[X1: X2 :...: Xk]
    Y=[Y1: Y2 :...: Yk]

    Construct an algebraic argument to show that
    X(X'X)^(-1)X'=Y(Y'Y)^(-1)Y'

    This is the idea I have:
    X=PYP^{-1}
    where P changes the basis from Y to X.

    Is this the right avenue?Thanks in advance.
     
  2. jcsd
  3. Oct 15, 2009 #2

    HallsofIvy

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    Didn't we just have this? Use the fact that (AB)-1= B-1A-1.
     
  4. Oct 15, 2009 #3
    Thanks for the help.

    There is a problem with the dimensions.

    X is n*k

    because each of X1,..,Xk is n*1

    So I was going to write (using your help)
    X(X'X)^{-1}X'= X(X^{-1}X'^{-1})X'
    But X^{-1} does not exist
     
    Last edited: Oct 15, 2009
  5. Oct 15, 2009 #4
    I think I need to use the fact that X(X'X)^(-1)X' is a projection operator somehow.
     
  6. Oct 15, 2009 #5

    HallsofIvy

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    If all columns in X are linearly independent- and you said they formed a basis for V1- then X must be non-singular and have an inverse!
     
  7. Oct 15, 2009 #6
    X is not a square matrix
     
  8. Oct 15, 2009 #7

    HallsofIvy

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    Ah! so A and B span V1, a subspace of V!
     
  9. Oct 15, 2009 #8
    yes

    any ideas?
     
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