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Math Help - linear transformation

  1. #1
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    linear transformation

    How do I find the matrix A to the linear transformation T: R^3 --> R^3

    it's defined by;

    1. reflection against 3x - 6y + 5z = 0

    then

    2. projection onto 2x + 6y + 4z = 0
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  2. #2
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    Here are the formulas that you need (I won't do the actual question for you).

    Suppose that \mathbf{n}=(a,b,c) is a unit vector (so that a^2+b^2+c^2=1). Then the projection onto the one-dimensional subspace spanned by n is P_{\mathbf{n}} = \begin{bmatrix}a^2&ab&ac\\ab&b^2&bc\\ac&bc&c^2\end  {bmatrix}.

    If px + qy + rz = 0 is the equation of a plane, let n be a unit vector orthogonal to the plane. So \textstyle\mathbf{n} = \frac1{\sqrt{p^2+q^2+r^2}}(p,q,r). Then the matrix of the projection onto the plane is I-P_{\mathbf{n}}, and the matrix of the reflection in the plane is I-2P_{\mathbf{n}}.

    To find the matrix for the composition of two such operations, form the matrices for each operation, then multiply them. So the matrix for reflection in 3x - 6y + 5z = 0 followed by projection onto 2x + 6y + 4z = 0 is (I-P_{\mathbf{n}})(I-2P_{\mathbf{m}}), where m and n are the normalised versions of (3,-6,5) and (2,6,4) respectively.
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  3. #3
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    I keep messing this one up, I've done it ten times and I still get the wrong answer.. Help?
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  4. #4
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    Unless I've also messed it up, you should get

    \textstyle\mathbf{m} = \frac1{\sqrt{70}}(3,-6,5),\quad \mathbf{n} = \frac1{\sqrt{56}}(2,6,4) = \frac1{\sqrt{14}}(1,3,2),

    P_\mathbf{m} = \frac1{70}\begin{bmatrix}9&-18&15\\ -18&36&-30\\ 15&-30&25\end{bmatrix},\qquad P_\mathbf{n} = \frac1{14}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix},

    I-2P_\mathbf{m} = \frac1{35}\begin{bmatrix}26&18&-15\\ 18&-1&30\\ -15&30&10\end{bmatrix},\qquad I-P_\mathbf{n} = \frac1{14}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix}.

    So the answer should be \frac1{35\times14}\begin{bmatrix}26&18&-15\\ 18&-1&30\\ -15&30&10\end{bmatrix}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix} (I'm not prepared to do the arithmetic to evaluate that).
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  5. #5
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    Quote Originally Posted by Opalg View Post
    Unless I've also messed it up, you should get

    \textstyle\mathbf{m} = \frac1{\sqrt{70}}(3,-6,5),\quad \mathbf{n} = \frac1{\sqrt{56}}(2,6,4) = \frac1{\sqrt{14}}(1,3,2),

    P_\mathbf{m} = \frac1{70}\begin{bmatrix}9&-18&15\\ -18&36&-30\\ 15&-30&25\end{bmatrix},\qquad P_\mathbf{n} = \frac1{14}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix},

    I-2P_\mathbf{m} = \frac1{35}\begin{bmatrix}26&18&-15\\ 18&-1&30\\ -15&30&10\end{bmatrix},\qquad I-P_\mathbf{n} = \frac1{14}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix}.

    So the answer should be \frac1{35\times14}\begin{bmatrix}26&18&-15\\ 18&-1&30\\ -15&30&10\end{bmatrix}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix} (I'm not prepared to do the arithmetic to evaluate that).
    Sorry.. wrong.. doh!
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