1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Transition matrix question.

  1. Jan 13, 2013 #1
    Hello,
    1. The problem statement, all variables and given/known data
    D:R3[x]->R3[x] is defined thus for any p(x)=(a0)+(a1)x+(a2)x2+(a3)x3:
    D(p(x)) = a1 + (2a2)x + (33)2
    I am asked to find [D]B where B is the standard basis {1,x,x2,x3}
    I am then asked to find the transition matrix from B to C, where C={1,1+x,x+x2,x2+x3}.
    Based on these two I am then asked to find [D]C.


    2. Relevant equations



    3. The attempt at a solution
    I have found [D]B to be (0 1 0 0)T (0 0 2 0)T (0 0 0 3)T (0 0 0 0)T
    I have found the transition matrix to be (1 -1 1 -1)T (0 1 -1 1)T (0 0 1 -1)T (0 0 0 1)T
    But then, the multiplication of (0 1 0 0)T (0 0 2 0)T (0 0 0 3)T (0 0 0 0)T by (1 -1 1 -1)T (0 1 -1 1)T (0 0 1 -1)T (0 0 0 1)T does not yield the expected (0 1 -1 1)T (0 0 2 -1)T (0 0 0 3)T (0 0 0 0)T
    Any ideas where I might be wrong? I have gone over the algebra several times, and have tried multiplying in the opposite order too.
     
  2. jcsd
  3. Jan 13, 2013 #2
    Hi there,

    The linear transformation D(p(x)) = a1 + (2a2)x + (33)2; is this correct, or should it read D(p(x)) = a1 + (2a2)x + (3a3)x2? If my assumption is correct, then I can help you with your question. If you are not mistaken, then I can't really help because I am unfamiliar with that notation.

    Anyways, I'll chip in assuming D(p(x)) = a1 + (2a2)x + (3a3)x2. I started by finding the transition matrix from the B basis to the C basis. When you say: (1 -1 1 -1)T (0 1 -1 1)T (0 0 1 -1)T (0 0 0 1)T, I guess you mean the transpose of (a b c d) where each (a b c d) is the column. I think you got the columns in opposite order. For the transition matrix, I got:

    \begin{bmatrix}
    1 & -1 & 1 & -1\\
    0 & 1 & -1 & 1 \\
    0 & 0 & 1 & -1 \\
    0 & 0 & 0 & 1
    \end{bmatrix}.

    When asked for [D]B, you're trying to find the matrix of the linear transformation, right? Again, this is assuming the linear transformation I wrote above: D(p(x)) = a1 + (2a2)x + (3a3)x2, but I get

    \begin{bmatrix}
    0 & 1 & -2 & -3\\
    0 & 0 & 2 & -3 \\
    0 & 0 & 0 & 3 \\
    0 & 0 & 0 & 0
    \end{bmatrix},

    since the coordinate vectors of the linear transformation of each member of the B basis with respect to the C basis are the above matrice's column vectors; i.e.,

    [D(1)]C =

    \begin{bmatrix}
    0\\
    0 \\
    0 \\
    0
    \end{bmatrix},

    [D(x)]C =

    \begin{bmatrix}
    1\\
    0 \\
    0 \\
    0
    \end{bmatrix},

    [D(x2)]C =

    \begin{bmatrix}
    -2\\
    2 \\
    0 \\
    0
    \end{bmatrix}, and

    [D(x3)]C =

    \begin{bmatrix}
    3\\
    -3 \\
    3 \\
    0
    \end{bmatrix}.

    I only have enough time to compute those two matrices. Either way let me know if you wrote the linear transformation correctly so I can determine if what I said was pointless or possibly helpful!
     
  4. Jan 14, 2013 #3
    Hi,
    I have managed to sort it out myself. Regardless, thank you!
    However, I have by now come across the following difficulty: finding T(u) where u = [-7 2 9]T, given T:R3->R3 a linear transformation satisfying:
    v1=[-2 1 4]T eignevector of T whose eigenvalue is p1=2
    AND
    v2=[3 0 -1]T eignevector of T whose eigenvalue is p2=5

    Attempt at solution:
    Clearly, T[-2 1 4]T = 2[-2 1 4]T = [-4 2 8]T and T[3 0 -1]T = 5[3 0 -1]T = [15 0 -5]T
    But I am not sure how to continue and derive T(u) based on the above.
    I'd appreciate your help.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Transition matrix question.
  1. Transition Matrix (Replies: 4)

  2. Transition Matrix (Replies: 3)

Loading...