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Rotation linear transformation

  1. Apr 3, 2014 #1
    1. The problem statement, all variables and given/known data

    Given below are three geometrically defined linear transformations from R3 to R3. You are asked to find the standard matrices of these linear transformations, and to find the images of some points or sets of points.
    a) T1 reflects through the yz-plane

    b) T2 projects orthogonally onto the xy-plane

    c) T3 rotates clockwise through an angle of 3π/4 radians about the z axis


    3. The attempt at a solution

    a)

    The standard matrix of T1 is

    -1,0,0
    0,1,0
    0,0,1

    b)


    The standard matrix of T2 is

    1,0,0
    0,1,0
    0,0,0

    c)

    The standard matrix of T3 is

    -sqrt(2)/2, sqrt(2)/2,0
    -sqrt(2)/2,-sqrt(2)/2,0
    0,0,1

    d)

    The image under T1 of the line segment joining the points (-2, -4, 3) and (2, 2, 4) is line segment joining the points

    (2,-4,3) and (-2,2,4)

    e)

    The point (-4, -4, -4) is first mapped by T2 and then T3. The coordinates of the resulting point are

    [1,0,0;0,1,0;0,0,0] [-sqrt(2)/2, sqrt(2)/2,0; -sqrt(2)/2,-sqrt(2)/2,0; 0,0,1] =
    [-sqrt(2)/2,0,0; 0,-sqrt(2)/2,0; 0,0,0]


    [-sqrt(2)/2,0,0; 0,-sqrt(2)/2,0; 0,0,0] [-sqrt(2),0,0;0,-sqrt(2)/2,0;0,0,0]

    =[0.5,0,0;0,0.5,0;0,0,0]

    (e) is wrong but why?
     
  2. jcsd
  3. Apr 6, 2014 #2

    BruceW

    User Avatar
    Homework Helper

    I agree with all your answers except in part c). You got to [-sqrt(2)/2,0,0; 0,-sqrt(2)/2,0; 0,0,0] which is the correct standard matrix for the operation T2 T3 (i.e. both operations, performed one after the other). And really, the question is asking you to use T3 T2, but luckily since T2 is diagonal, the order doesn't matter here (but generally it will). So anyway, they are asking you to apply [-sqrt(2)/2,0,0; 0,-sqrt(2)/2,0; 0,0,0] to a point. But you seem to apply this matrix to itself for some reason... They are asking you to apply the matrix to a point, not to itself...
     
  4. Apr 7, 2014 #3
    It could be a careless mistake. I'll review it.
     
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