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Finding the matrix of a transformation

  1. Oct 20, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider the transformation T from R2 to R2 that rotates any vector x through an angle of 45 degrees in the counterclockwise direction. You are told that T is a linear transformation. Find the matrix of T.


    2. Relevant equations



    3. The attempt at a solution

    A vector with components x1 and x2 should become x 2 and x1, seeing that the transformation should rotate the vector 45 degrees. So therefore, the matrix of the transformation should be:

    [0 1]
    [1 0]
     
  2. jcsd
  3. Oct 20, 2011 #2

    Mark44

    Staff: Mentor

    The rotation you're describing here is a rotation of 90° counterclockwise, not 45°.

    What should T do to x1 = <1, 0> and x2 = <0, 1>?
     
  4. Oct 20, 2011 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    No, with a 45 degree rotation, any vector on the x-axis (y=0) would rotate to the line y= x while any vector on the y-axis would rotate to the line y= -x. What vectors on those lines have length 1?
     
  5. Oct 20, 2011 #4
    It should move it half of 90 degrees. But what matrix would achieve that? A matrix that looks like this perhaps:

    [0 1/2]
    [1/2 0]

    How should you think when you have problems of this sort?
     
  6. Oct 20, 2011 #5

    Mark44

    Staff: Mentor

    No, the matrix above doesn't work.

    I'll ask this again.
     
  7. Oct 20, 2011 #6

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Which (because I just have to jump in behind Mark44) is much what I asked before: what vector, <x, x>, has length 1? What vector <-x, x>, has length 1?
     
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