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Math Help - Find the standard matrix of an orthogonal projection

  1. #1
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    Find the standard matrix of an orthogonal projection

    Problem 4. Let W = Span[w1; w2], where
    w1 = [3 0 4]
    w2 = [4 5 -3]

    If T (from R^3 to R^3) is the orthogonal projection onto W, find the standard matrix of T.

    I did this by finding the orthogonal projections of e1, e2, and e3 on span W and constructing the transformation from [e1 e2 e3]

    My answer was :
    0.680 0.400 0.240
    0.400 0.500 -0.300
    0.240 -0.300 0.820

    Is this correct? I double checked by finding that the applying the transformation matrix to w1 and w2 returned w1 and w2 which is what it should do. Thanks!
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  2. #2
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    Quote Originally Posted by eiad77 View Post
    Problem 4. Let W = Span[w1; w2], where
    w1 = [3 0 4]
    w2 = [4 5 -3]

    If T (from R^3 to R^3) is the orthogonal projection onto W, find the standard matrix of T.

    I did this by finding the orthogonal projections of e1, e2, and e3 on span W and constructing the transformation from [e1 e2 e3]

    My answer was :
    0.680 0.400 0.240
    0.400 0.500 -0.300
    0.240 -0.300 0.820

    Is this correct? I double checked by finding that the applying the transformation matrix to w1 and w2 returned w1 and w2 which is what it should do. Thanks!
    Alternatively, you can use the fact that w_1/5 and w_2/\sqrt{50} form an orthonormal basis for W to see that T is given by the formula Tx = \tfrac1{25}\langle x,w_1\rangle w_1 + \tfrac1{50}\langle x,w_2\rangle w_2. The resulting matrix agrees with the one above.
    Last edited by Opalg; April 24th 2010 at 07:31 AM.
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