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Transition Matrix of T

  1. Jun 25, 2017 #1
    1. The problem statement, all variables and given/known data
    Find the transition matrix ##P## of a transformation defined as
    ##T:ℝ_2→ℝ_3##
    ##T:\begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix}a+2b\\-a\\b\end{bmatrix}##

    For basis

    ##B=\begin{bmatrix}1\\2\end{bmatrix},\begin{bmatrix}3\\-1\end{bmatrix}##

    ##C=\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}1\\1\\1\end{bmatrix}##

    Verify this matrix by calculating ##Pv## and comparing the result with the actual transformation ##T(v)## for

    ##v=\begin{bmatrix}-7\\7\end{bmatrix}_B##

    2. Relevant equations

    ##[x]_C=P_C←_B[x]_B##

    3. The attempt at a solution

    When applying ##T## the results shows

    ##T(v)=\begin{bmatrix}7\\7\\7\end{bmatrix}##

    However, my attempts to find the transition matrix ##P## have been unsuccessful, apparently my struggle is with the fact that ##ℝ_2## maps to ##ℝ_3##, therefore the transition matrix ##P## has to be ##3x2##, thus non invertible, so I can't find it through its inverse, I attempted separating the transformation into its components like

    ##T(v)= a \begin{bmatrix}1\\-1\\0\end{bmatrix} + b\begin{bmatrix}2\\0\\1\end{bmatrix}##

    Or ##T(B_1), T(B_2) ##

    ##T(B_1)= \begin{bmatrix}5\\-1\\2\end{bmatrix}##
    ##T(B_2)= \begin{bmatrix}1\\-3\\-1\end{bmatrix}##

    But neither ## \begin{bmatrix}1&2\\-1&0\\0&1\end{bmatrix}## nor ##\begin{bmatrix}5&1\\-1&-3\\2&-1\end{bmatrix}## are the same as on my answer key therefore I believe my approach to this matrix is flawed.

    Any advise would be appreciated.
     
  2. jcsd
  3. Jun 26, 2017 #2

    WWGD

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    Usually transition matrix is between different spaces for the same vector space. How are you defining it?
     
  4. Jun 26, 2017 #3

    haruspex

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    I agree with WWGD that normally "transition matrix" means a mapping from and to the same vector space. Putting that aside...

    I did not follow your method. I would write B as (b1, b2) etc. then see how to write ##\begin{bmatrix}a\\b\end{bmatrix}## as a linear combination of the vectors b1, b2. Similarly, how to write ##\begin{bmatrix}a+2b\\-a\\b\end{bmatrix}## in terms of the ci basis vectors.
     
  5. Jun 27, 2017 #4
    Thank you both.

    Indeed my problem is that I was missing the step to write the vectors in terms of Ci basis.

    However I've found I have a conceptual mistake in my question. Apparently, a transition matrix is completely different from a matrix associated to a transformation, and it was the later the one I was looking for. How different are these two?
     
    Last edited: Jun 27, 2017
  6. Jun 27, 2017 #5

    haruspex

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    Transition suggests a change of state within a system. In a vector space context that would mean a transition from one state vector to another state vector within the same space.
     
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