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Find transformation

  1. Jun 15, 2011 #1
    I'm not exactly sure how to find the transformation. The professor wrote something different in class. I know [T]α is what you multiply with the "new" basis to get the transformation of the components of the "original" basis. In this case, it's simply still alpha.

    http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110615_211957.jpg [Broken]

    http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110615_204346.jpg [Broken]
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Jun 15, 2011 #2

    lanedance

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    so guessing here, and abusing a little notation but hopefully it helps..

    for a given matrix A you should able to write in the alpha basis:
    [tex] A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
    = q\vec{\alpha}_1+pq\vec{\alpha}_2+rq\vec{\alpha}_3+sq\vec{\alpha}_3 = \begin{pmatrix} p \\ q \\ r \\ s \end{pmatrix}_{\alpha} [/tex]

    then apply the T transform which is already written in the alpha basis
     
  4. Jun 15, 2011 #3

    lanedance

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    to further understand the alpha basis, note that you could consider A expressed in the standard basis, call it s, and write
    [tex] A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
    = a\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}
    +b\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}
    +c\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}
    +d\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix}_s [/tex]
     
  5. Jun 15, 2011 #4

    lanedance

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    updated above
     
  6. Jun 15, 2011 #5
    Oh, I see what you're doing.
     
  7. Jun 15, 2011 #6

    lanedance

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    what's not the alpha basis?

    you need to solve for q,p,r,s which give the components in the alpha basis
     
  8. Jun 15, 2011 #7
    The components are already given.
     
  9. Jun 16, 2011 #8

    lanedance

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    the way i read it (open to interp):
    - the components of A in the standard basis are given
    - the components of the operator T in the alpha basis is given

    so i think you need to express A in the alpha basis, or express T in the standard basis
     
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