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Finding transition matrices.

  1. May 12, 2012 #1
    1. The problem statement, all variables and given/known data
    Let V be the vector space of all symmetric 2x2 matrices, and consider the bases.
    S = {
    [1 0] [0 1] [0 0]
    [0 0],[1 0],[0 1]}

    B = {
    [1 1] [-1 1] [1 0]
    [1 2],[ 1 1],[0 1]}
    of V.
    Find the transition matrix Ps,b. Use your answer to calculate Pb,s.

    2. Relevant equations
    a = [itex]\alpha[/itex][itex]_{1}[/itex]b[itex]_{1}[/itex] + [itex]\alpha[/itex][itex]_{2}[/itex]b[itex]_{2}[/itex] + [itex]\alpha[/itex][itex]_{3}[/itex]b[itex]_{3}[/itex] +...... + [itex]\alpha[/itex][itex]_{k}[/itex]b[itex]_{k}[/itex]


    3. The attempt at a solution
    I honestly don't know where to start. All previous questions like this we've dealt with vectors and not 2x2 matrices..

    ie B = (1,3),(2,1)

    (1,3) = 1(1,0) + 3(0,1) and (2,1) = 2(1,0) + 1(0,1)

    So Ps,b = [1 2] and Pb,s is just the inverse of Ps,b = -1/5[1 -2]
    [3 1] [-3 1]

    But I don't know how to even start when I'm given 2x2 matrices..
     
  2. jcsd
  3. May 13, 2012 #2
    haha im doing the same linear assignment. Thought id be awesome and let you know that Ps,b is {(1, -1, 1), (1, 1, 0), (2, 1, 1)}. make an identity matrix to solve the inverse to get Pb,s.
     
  4. May 13, 2012 #3
    Haha yeah, I ended up figuring it out - I never usually get a response from this website, I don't know why, yours is the first one I've gotten so thanks.

    Let me know if you need a hand with either of the other questions.
     
  5. May 13, 2012 #4
    No worries mate, and actually im pretty good for the other two, was just a bit unsure of Q1 b). did you manage to sus that?
     
  6. May 13, 2012 #5
    For 1b) I got
    [ 1 2 -1] [-1] = [0]
    [-1 -1 1] [ 3] [3]
    [-1 -3 2] [ 5] [2]

    That's written out pretty garbagety, but hopefully you can decipher it.. it's Pb,s multiplied by (-1,3,5)
     
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