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Math Help - ordered bases

  1. #1
    Member Jskid's Avatar
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    ordered bases

    Let S={\vec v_1, \vec v_2} and T={\vec w_1, \vec w_2} be bases for P_1 where \vec w_1 = t-1 and \vec w_2 = t+1. If the transition matrix from T to S is \left[ {\begin{array}{cc}<br />
 1 & 2  \\<br />
 2 & 3  \\<br />
 \end{array} } \right]<br />
determine S.

    So the columns of the matrix are the coordinates fo the T-basis vectors with respect to the S-basis, but that's not good because S is what we're trying to find
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    There is a little problem about terminology. I don't know if your teaher means:


    (i)\;[p(t)]_S=\begin{bmatrix}{1}&{2}\\{2}&{3}\end{bmatrix}[p(t)]_T\textrm{\;\;or\;\;}(ii)\;[p(t)]_T=\begin{bmatrix}{1}&{2}\\{2}&{3}\end{bmatrix}[p(t)]_S

    Supposing (i) we have:

      [\vec{w_1}]_T=\begin{bmatrix}{1}\\{0}\end{bmatrix},\; [\vec{w_2}]_T=\begin{bmatrix}{0}\\{1}\end{bmatrix}

    So,

    [\vec{w_1}]_S=\begin{bmatrix}{1}&{2}\\{2}&{3}\end{bmatrix}\be  gin{bmatrix}{1}\\{0}\end{bmatrix}=\begin{bmatrix}{  1}\\{2}\end{bmatrix}

    Then, \vec{w_1}=t-1=\vec{v_1}+2\vec{v_2} etc .
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