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Math Help - Matrix transformations

  1. #1
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    Matrix transformations

    What matrices represent these transformations with respect to the standard basis?
    T: R^3 -----> R^3

    1) T rotates each point through an angle theta about the x-axis in the direction that moves the point (0,0,1) towards the point (0,-1,0)

    2) T reflects each point in the (x,z) - plane
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  2. #2
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    Quote Originally Posted by maximus101 View Post
    What matrices represent these transformations with respect to the standard basis?
    T: R^3 -----> R^3

    1) T rotates each point through an angle theta about the x-axis in the direction that moves the point (0,0,1) towards the point (0,-1,0)

    2) T reflects each point in the (x,z) - plane
    Remember that you just need to find the transformation of the basis vectors in \mathbb{R}^3

    For 1) this is just a counterclockwise rotation in the y,z plane. It will fix any point on the x axis. Now just draw a 2d picture to Find out what
    T(e_2) and T(e_3) are.
    Hint: the transform will not change the length of the vector it will stay on the unit circle Now just convert from polar back to rectangular coordinates.
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  3. #3
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    To find the matrix representation of a linear transformation in a given basis, apply the transformation to each basis vector in turn and write the result as a linear combination of the basis vectors. The coefficients give the columns of the matrix.

    For example, the "reflection in the xz-plane" maps each point (x,y,z) into (x,-y,z) so maps the basis vectors (x, 0, 0)-> (x, 0, 0), (0, y, 0)-> (0 -y, 0), (0, 0, z)-> (0, 0, z).
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