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Linear Algebra - Matrix Transformations

  1. Mar 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Let L denote the line through the origin in R2 that that makes angle -∏ < theta ≤ ∏
    with the positive x-axis. The reflection operator that reflects points about L in R2
    is the matrix transformation R2 --> R2 with standard matrix

    [cos 2(theta) sin 2(theta); sin 2(theta) -cos 2(theta)]

    Show that the composition of a rotation operator followed by a re
    reflection operator is another reflection operator.

    2. Relevant equations

    Standard matrix for reflection in R2 comes to mind;

    [cos(theta) -sin(theta); sin (theta) cos (theta)]

    These are 2x2 matrices and I'm not sure how to input them here so I put ; to seperate the two columns.

    3. The attempt at a solution

    I'm kind of confused as to how to attempt to solve the question. I mean I could use the 2 standard matrices and use matrix multiplication which gives a really ugly 2x2 that I'm not sure what to do with. Any help to get me started would be appreciated :).
     
    Last edited: Mar 25, 2012
  2. jcsd
  3. Mar 25, 2012 #2
    Anybody? : (
     
  4. Mar 26, 2012 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    It's pretty straight forward isn't it? You know how to write the rotation and reflection as matrices. The "rotation followed by a reflection" is the product of those two matrices. Do that multiplication and show that it is a reflection by determining the appropriate angle for the "line of reflection"?
     
  5. Mar 26, 2012 #4
    Thanks HallsofIvy! It turns out that I read the question wrong. I just had to use matrix multiplication and simplify. When I did it yesterday, I wasn't sure about how to simplify it. But using trig identities makes it very simple. Thanks again for your help!
     
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