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Math Help - Linear transformations!

  1. #1
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    Linear transformations!



    need some help getting started. I know how to project, rotate, reflect.
    But im stuck anyways


    Thanks!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Perhaps you meant v -> v x u instead of u -> v x u . If so, find first

    v x u = ( v_1 , v_2 , v_3 ) x ( -3 , 2 , 3 ) = ...

    and apply the reflection to v x u .
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  3. #3
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    oh, its v=(-3,2,3) not u.
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  4. #4
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    Thanks fernando for the help.

    So i did it this way:




    But its not the correct answer. What did i do wrong?
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by mechaniac View Post
    But its not the correct answer. What did i do wrong?

    If u = ( x , y , z ) then

    T_1 ( x , y . z ) = ( -3 , 2 , 3 ) x ( x , y , z ) =

    ... = ( 2 z - 3 y , 3 z + 3 x , - 3 y - 2 x )
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  6. #6
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    oh i did a misstake with the crossproduct. There is a typo in the T2 matrix to but no worries, got it now! Big thanks
    Last edited by mechaniac; April 20th 2011 at 08:19 AM.
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  7. #7
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    The simplest way to find a matrix representation of a linear transformation is to look at what the transformation does to the basis vectors. For example, to map <1, 0, 0> onto <1, 0, 0> X <-3, 2, 3> means to map <1, 0, 0> onto <0, -3, -3>. To map <0, 1, 0> onto <0, 1, 0> X <-3, 2, 3> means to map <0, 1, 0> onto <3, 0, 3>. To map <0, 0, 1> onto <0, 0, 1> X <-3, 2, 3> means to map <0, 0, 1> onto <-2, -3, 0>. The matrix giving that is
    [ 0 3 -2]
    [ -3 0 -3]
    [ -3 3 0]
    the matrix having those results as columns. To see that, mutiply that matrix by <1, 0, 0>, <0, 1, 0>, and <0, 0, 1> succesivly.

    Reflection in x= z maps <1,0, 0> into <0, 0, 1> and <0, 0, 1> into <1, 0, 0>. Since <0, 1, 0> is in that plane, it is mapped into itself.

    The matrix giving that transformation is
    [ 0 0 1]
    [0 1 0]
    [1 0 0]

    To find the matrix representing both of those transformations, multiply them (in the proper order, of course).
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