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Finding Transition Matrix

  1. Apr 14, 2014 #1
    1. The problem statement, all variables and given/known data
    let a= {a1, a2, a3, a4}
    and b={b1,......,b4}
    and r = {r1,...,r4}

    Also,
    b1 = 4a1
    b2 = 8a1 + 7a2
    b3 = 4a1 + 4a2 + 4a3
    b4 = 9a1 + 5a2 + 8a3 + 5a4

    and

    r1 = 3b4
    r2= 4b3 + 6b4
    r3 = 9b2 + 3b3 + 9b4
    r4 = 6b1 + 5b2 + 3b3 + 5b4

    Find transition matrix from basis a to r.


    2. Relevant equations

    To find transition matrix I know you make matrix with on one side one basis and the other side the other basis and turn one of them into the identity matrix.
    Hard to visually represent this equation, but hopefully you know what I mean.


    3. The attempt at a solution

    Normally these problems are really easy in class with bases like {(1,2), (2,1)}
    I have no idea how to solve this one.
    thus far I have but r in terms of a:

    r1 = 27a1 + 15a2 + 24a3 + 15a4
    r2 = 70a1 + 46a2 + 64a3 + 30a4
    r3 = 165a1 + 120a2 + 84a3 + 45a4
    r4 = 121a1 + 72a2 + 40a3 + 25a4

    What do I do from here?
     
  2. jcsd
  3. Apr 14, 2014 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    You've done it; now just write these equations in matrix form. That is, fill in the blanks in
    [tex] \pmatrix{r_1\\r_2\\r_3\\r_4} =
    \pmatrix{* & * & * & * \\ * & * & * & * \\ * & * & * & * \\ * & * & * & * }
    \pmatrix{a_1 \\a_2 \\a_3 \\a_4}[/tex]
     
    Last edited: Apr 14, 2014
  4. Apr 14, 2014 #3
    Okay, I still don't understand. Do I put the coefficients of r and a and solve for whatever is in the matrix of *?

    What are the coefficients of a? All ones? I'm confused
     
  5. Apr 14, 2014 #4

    Ray Vickson

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    Science Advisor
    Homework Helper

    It sounds like you are saying that you do not understand vectors, matrices and matrix multiplication. You need to go to the library and take and read out a book on the subject; it is much too lengthy to be presented in a help forum. You can also try on-line notes and tutorials on the subject.

    Please note: I am NOT trying to insult you; it really does sound to me as though you do not have the background to tackle the question.
     
  6. Apr 15, 2014 #5
    no i get that the matrix with the astrisk's should contain the coefficients of r. What are the components of the matrix with a1 through a4?
     
  7. Apr 15, 2014 #6
    Wait hold on I think I just realized how to solve this problem. You can get one transition matrix from another transition matrix by taking the inverse right? So can I just take inverse of the matrix with the asterisks?
     
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