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Math Help - Change of Basis

  1. #1
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    Change of Basis

    Let  V \subset R^3 be the plane defined by the equation  2x_1 + x_2 =0.  Let T:R^3 \rightarrow R^3 be the linear transformation defined by reflecting across V. Find the standard matrix for T.

    This should be done using change of basis formula. The standard matrix for T, A, should be  A=PBP^{-1} , where B is the matrix of the transformation with respect to some basis. I chose my basis to be two vectors that are a basis of V and a vector that is in V perp.

    How do I find B and solve this problem?
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  2. #2
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    Quote Originally Posted by BrianMath View Post
    Let  V \subset R^3 be the plane defined by the equation  2x_1 + x_2 =0.  Let T:R^3 \rightarrow R^3 be the linear transformation defined by reflecting across V. Find the standard matrix for T.

    This should be done using change of basis formula. The standard matrix for T, A, should be  A=PBP^{-1} , where B is the matrix of the transformation with respect to some basis. I chose my basis to be two vectors that are a basis of V and a vector that is in V perp.

    How do I find B and solve this problem?
    You have made a good choice of the basis. The operation of reflection across V will leave vectors in V unchanged, and it will take a vector in V perp to its negative. So the matrix B should be \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{bmatrix}.
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  3. #3
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    I tried doing this myself where B is  \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{bmatrix} and P is  \begin{bmatrix}1&0&2\\ -2&0&1\\ 0&1&0\end{bmatrix} where the first two columns of P are a basis of the plane V, and the third column is in V perp, however, my answer did not match the book's answer. Is my choice of P wrong?
    Last edited by BrianMath; December 2nd 2010 at 07:46 AM. Reason: typed the matrix in wrong
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  4. #4
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    oh wait, nevermind!! I was just looking at the wrong solution at the back of the book. HAHA my bad. But this means I understand it more then I thought I did!!
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