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Linear algebra; changing bases

  1. Dec 24, 2007 #1
    [SOLVED] Linear algebra; changing bases

    1. The problem statement, all variables and given/known data
    I have two bases E and F given by:

    E = [1, x, x^2]

    F = [1-x, x-x^2, x^2].

    I want to find the transition-matrix S that goes from E to F.

    3. The attempt at a solution
    To do this, I must write one basis as a linear combination of the other. I am just confused about which way?

    Since we are going from E to F, I believe I have to write F as l.c. of E, so for 1 - x we have the vector (1,-1,0)^T. Am I right or is it the other way around?

    I hope you can help, thanks in advance.
     
  2. jcsd
  3. Dec 24, 2007 #2
    Right idea, wrong way. You want to take an arbitrary coordinate vector (x, y, z) written in terms of the basis E to the corresponding coordinate vector in terms of basis F. But the easiest way to do this is to just find the transformation of (1, 0, 0), (0, 1, 0), and (0, 0, 1). That is, find the elements of E in terms of F.
     
  4. Dec 25, 2007 #3
    I see, thank you.
     
  5. Dec 25, 2007 #4
    Although, I have a new question, if you do not mind.


    I have 4 matrices that span out a vector-space W. The 4 matrices are 2x2 matrices, and they are:

    A_1 = (1 0 , 0 0) - (that is 1 0 in top, 0 0 in bottom).

    A_2 = (0 1 , 0 0)

    A_3 = (0 0 , 1 0)

    A_4 = (0 0 , 0 1).

    We have another matrix A = (a b , c d) and a linear transformation F : W -> W given by:

    F(X) = AX-XA, X in W.

    I have to find the matrix for F with respect to the basis W spanned by A_1 .. A_4.

    _____

    What I did was to find F(A_1) up to F(A_4) and then express this result as a linear combination of A_1 to A_4, e.g.:

    F(A_1) = 0*A_1 - b*A_2 + c*A_3 - 0*A_4. Then (0,-b,c,0)^T is the first column in my matrix. Is this approach correct?
     
  6. Dec 25, 2007 #5
    That's right, don't forget to write the linear combinations as column vectors in your transition matrix.
     
  7. Dec 25, 2007 #6
    Thanks. Do you have any comments/suggestion to my second question?
     
  8. Dec 26, 2007 #7
    About my last question - the reason why I don't think it's correct is that I get a 4x4-matrix, but the vector-space is spanned by 4 2x2 matrices?
     
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