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Homework Help: Inverse of a linear transformation

  1. Mar 22, 2006 #1
    find the inverse of [tex] T \left[ \begin{array}{cc} a&b \\ c&d \end{array} \right] = \left[ \begin{array}{cc} a+2c&b+2d \\ 3c-a&3d-b \end{array} \right] [/tex]

    do i row reduce the transformation matrix... it doesn work , though
    is there an easier way??
     
  2. jcsd
  3. Mar 22, 2006 #2

    0rthodontist

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    You have to recognize that writing a b c d as a matrix like that is similar to writing a b c d as a column vector. The _real_ matrix of the transformation is 4x4.
     
  4. Mar 23, 2006 #3

    HallsofIvy

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    As Orthodontists pointed out (remarkable how good at math a dentist is!) since this maps a 2 by 2 matrix into a 2 by 2 matrix, the actual "transformation matrix" is 4 by 4. If you don't want to write all of that out, try a more basic approach. Using p,q,r,s as the components of the result of the transformation, you have four equations:
    p= a+ 2c, q= b+ 2d, r= 3c- a, s= 3d- b, four equation in the four "unknowns" a, b, c, d. Solve for a, b, c, d in terms of p,q,r,s. that will give the equations for the inverse transformation.
     
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