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!

Homework Help: 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
    B={(1,2,1),(0,1,-1),(2,3,2)}

    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

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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

    Redbelly98

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    Moderator's note:
    Homework assignments or textbook style exercises for which you are seeking assistance are to be posted in the appropriate forum in our https://www.physicsforums.com/forumdisplay.php?f=152" area. This should be done whether the problem is part of one's assigned coursework or just independent study.
     
    Last edited by a moderator: Apr 24, 2017
  6. Feb 16, 2010 #5

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

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

    [tex]
    \vec{x}=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1=\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2
    \hspace{0.5in}
    \vec{y}=\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}_1=\begin{pmatrix}y_1'\\y_2'\\y_3'\end{pmatrix}_2[/tex]

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

    [tex]\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1=P\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2[/tex]

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

    [tex]\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2=P^{-1}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1[/tex]

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

    [tex]\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}_1=A\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1
    \hspace{0.5in}
    \begin{pmatrix}y_1'\\y_2'\\y_3'\end{pmatrix}_2=B\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2
    [/tex].

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

    [tex]\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1=P\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2[/tex]

    [tex]\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}_1=A\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}_1=AP\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2[/tex]

    [tex]\begin{pmatrix}y_1'\\y_2'\\y_3'\end{pmatrix}_2=P^{-1}\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}_1=P^{-1}AP\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}_2[/tex]

    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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook