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Math Help - Orthogonal Projections and Linear Transformations

  1. #1
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    Orthogonal Projections and Linear Transformations

    Much help would be appreciated on these problems, thanks!

    1.)
    In R^3 the orthogonal projections on the x-axis, y-axis and z-axis are
    defined by T1(x, y, x) = (x, 0, 0), T2(x, y, z) = (0, y, 0), and T3(x, y, z) = (0, 0, z)
    respectively.
    (a) Show that the orthogonal projections on the coordinate axes are linear
    operators, and find their standard matrices.
    (b) Show that if T: R^3 R^3 is an orthogonal projection on one of the coordinate
    axes, then for every vector x in R^3, the vector T(x) and x T(x) are
    orthogonal vectors.
    (c) Make a sketch showing x and x - T(x) in the case where T is the orthogonal
    projection on the x-axis.


    2.)
    (a)
    Is a composition of one-to-one linear transformations one-to-one? Justify
    your conclusion.
    (b) Can the composition of a one-to-one linear transformation and a linear
    transformation that is not one-to-one be one-to-one? Account for both possible
    orders of composition and justify your conclusion.
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  2. #2
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    Quote Originally Posted by eddie25 View Post
    Much help would be appreciated on these problems, thanks!

    1.)
    In R^3 the orthogonal projections on the x-axis, y-axis and z-axis are
    defined by T1(x, y, x) = (x, 0, 0), T2(x, y, z) = (0, y, 0), and T3(x, y, z) = (0, 0, z)
    respectively.
    (a) Show that the orthogonal projections on the coordinate axes are linear
    operators, and find their standard matrices.
    (b) Show that if T: R^3 R^3 is an orthogonal projection on one of the coordinate
    axes, then for every vector x in R^3, the vector T(x) and x T(x) are
    orthogonal vectors.
    (c) Make a sketch showing x and x - T(x) in the case where T is the orthogonal
    projection on the x-axis.


    2.)
    (a) Is a composition of one-to-one linear transformations one-to-one? Justify
    your conclusion.
    (b) Can the composition of a one-to-one linear transformation and a linear
    transformation that is not one-to-one be one-to-one? Account for both possible
    orders of composition and justify your conclusion.

    I bet you've already done some self work on both these problems, so show it to us and tell us where're you stuck and somebody will perhaps try to help you.

    Tonio
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  3. #3
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    Honestly, I don't know 1.) very much.

    2.) I know for a.) I used:

    Ta(Ta^(-1)(x)) = AA^(-1)x = /x = x on two one-to-one linear transformations. I used a:

    X 0
    0 X

    matrix to see if I get the same equation back. Same principle for b.)
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