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Transition Matrix

  1. Feb 16, 2010 #1
    I came across this problem in one of my linear algebra books.
    A linear transformation T:R^3 ->R^3 has matrix

    2 3 0
    -1 1 2
    2 0 1
    with respect to the standard basis for R^3. Find the matrix of T with respect to the basis

    The answer given is
    -28 -19 -43
    5 4 7
    18 11 28
    but i have no idea how to get to that answer as the book does not provide workings/steps. Any help would be appreciated thanks.
  2. jcsd
  3. Feb 16, 2010 #2


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    I recommend that you read this post (the part above the quote) to make sure that you understand the relationship between linear operators and matrices.
  4. Feb 16, 2010 #3
    I'll use subscript B to indicate a vector or linear transformation expressed as a matrix in the new basis, thus

    [tex]\left ( Tx \right )_B = T_B x_B[/tex]

    Let B be a matrix whose columns are the basis vectors of the new basis, expressed in the standard basis. Then the inverse of B will convert the components of a general vector from the standard basis to the new basis:

    [tex]B^{-1}Tx = T_B B^{-1}x.[/tex]

    So [itex]B^{-1}T[/itex] has the same effect on [itex]x[/itex] as [itex]T_B B^{-1}[/itex]. Now all we have to do is solve for [itex]T_B[/itex].

    [tex]B^{-1}T = T_B B^{-1}[/itex]

    [tex]B^{-1}TB = T_B.[/itex]
  5. Feb 16, 2010 #4


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    Last edited by a moderator: Apr 24, 2017
  6. Feb 16, 2010 #5


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    Say vectors [itex]\vec{x}[/itex] and [itex]\vec{y}[/itex] have the representations


    with respect to basis 1 and basis 2. You can construct a matrix P that will convert between the two representations:


    and its inverse P-1 will take you in the other direction:


    If [itex]\vec{y}=T(\vec{x})[/itex], there are matrices A and B such that


    It turns out that A and B are related by [itex]B=P^{-1}AP[/itex] because




    In your problem, you're given A, and you want to find B. So the problem boils down to finding P given the information you have about the two bases.
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