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Math Help - Change of basis

  1. #1
    MHF Contributor arbolis's Avatar
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    Change of basis

    In a given basis \{ \vec e _i \} of a vector space, a linear transformation and a given vector of this space are determinate by \begin{bmatrix}  2 & 1 & 0 \\ 1&2&0 \\ 0 & 0 & 5 \end{bmatrix} and \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} (should be a column vector).
    Find the matrix representation of the transformation and of the vector in a new basis such that the old one is represented by \vec e _1 = \begin{bmatrix} 1&1&0 \end{bmatrix}, \vec e _2 = \begin{bmatrix} 1&-1&0 \end{bmatrix}, \vec e _3 = \begin{bmatrix} 0&0&1 \end{bmatrix} (they should be column vectors).

    My attempt: I formed a matrix whose columns are the \vec e_1 , \vec e_2 , \vec e_3 and I found its inverse. Call them S and S^{-1}.
    Then I multiplied S^{-1}AS where A is the first matrix I wrote. As result, I obtained the following matrix A'= \begin{bmatrix} 3&0&0 \\ 0&1&0 \\ 0&0&5  \end{bmatrix} which would be the matrix they asked for.

    For the vector, I wasn't sure at all... I just multipled A' by \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} and I obtained \begin{bmatrix} 3 & 12 & 15 \end{bmatrix} which would be the vector they ask for.
    Am I right?
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    Quote Originally Posted by arbolis View Post
    In a given basis \{ \vec e _i \} of a vector space, a linear transformation and a given vector of this space are determinate by \begin{bmatrix}  2 & 1 & 0 \\ 1&2&0 \\ 0 & 0 & 5 \end{bmatrix} and \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} (should be a column vector).
    Find the matrix representation of the transformation and of the vector in a new basis such that the old one is represented by \vec e _1 = \begin{bmatrix} 1&1&0 \end{bmatrix}, \vec e _2 = \begin{bmatrix} 1&-1&0 \end{bmatrix}, \vec e _3 = \begin{bmatrix} 0&0&1 \end{bmatrix} (they should be column vectors).

    My attempt: I formed a matrix whose columns are the \vec e_1 , \vec e_2 , \vec e_3 and I found its inverse. Call them S and S^{-1}.
    Then I multiplied S^{-1}AS where A is the first matrix I wrote. As result, I obtained the following matrix A'= \begin{bmatrix} 3&0&0 \\ 0&1&0 \\ 0&0&5  \end{bmatrix} which would be the matrix they asked for.

    For the vector, I wasn't sure at all... I just multipled A' by \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} and I obtained \begin{bmatrix} 3 & 12 & 15 \end{bmatrix} which would be the vector they ask for.
    Am I right?
    The first part, A', is correct but the question did not ask for the vector <1 2 3> multiplied by A'.

    Suppose your problem were Av= u where [tex]A= \begin{barray}2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 5\end{bmatrix} and [tex]v= \begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}.

    To change A to A' you would, as you say, write S^{-1}AS with the S matrix you give. Now think how we would apply that to the equation Av= u. If we multiply both sides, on the left, by S^{-1}, we would have S^{-1}Av= S^{-1}u. But that does not have the "S" multiplying A on the right. Just multiplying the equation on the right won't help because that would give S^{-1}AvS= S^{-1}uS, not what we want.

    Instead, use the fact that I= SS^{-1} to write the equation as S^{-1}A(SS^{-1})v= S^{-1}u or (S^{-1}AS)(S^{-1}v)= S^{-1}u- that is, A'v'= u' with A'= S^{-1}AS and v'= S^{-1}v, not A'v.
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