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Thread: Orthonormal basis

  1. #1
    Junior Member
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    Orthonormal basis

    let$\displaystyle (e_{1},e_{2},e_{3})$ be a positive oriented orthonormal basis.

    Define a new basis $\displaystyle (f_{1},f_{2},f_{3})$

    * Show that $\displaystyle (f_{1},f_{2},f_{3})$ also is a orthonormal basis.
    * determine the coordinates for the vector $\displaystyle u = 7e_{1}+1e_{2}-4e_{3}$ in the base $\displaystyle (f_{1},f_{2},f_{3})$

    and:
    $\displaystyle f_{1}= \frac{1}{3}e_{1} - \frac{2}{3}e_{2} + \frac{2}{3}e_{3} $

    $\displaystyle f_{2}= \frac{2}{3}e_{1} + \frac{2}{3}e_{2} + \frac{1}{3}e_{3} $

    $\displaystyle f_{3}= -\frac{2}{3}e_{1} + \frac{1}{3}e_{2} + \frac{2}{3}e_{3} $

    i thought this would be the coordinates for the vector u:

    $\displaystyle
    1/3\[ \left( \begin{array}{ccc}
    1 & -2 & 2 \\
    2 & 2 & 1 \\
    -2 & 1 & 2 \end{array} \right)\]
    $$\displaystyle \[ \left( \begin{array}{ccc}
    7 \\
    1 \\
    -4 \end{array} \right)\] $=$\displaystyle \[ \left( \begin{array}{ccc}
    -1 \\
    4 \\
    -7 \end{array} \right)\] $

    $\displaystyle u=(-1,3,-7)$

    I need some advice for the first problem on how to show that $\displaystyle (f_{1},f_{2},f_{3})$ is an orthonormal basis to.

    Thanks!



    Edit: dont need help to show that $\displaystyle (f_{1},f_{2},f_{3})$ is an orthonormal basis. Figured it out right after i posted
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  2. #2
    MHF Contributor

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    Tejas
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    your calculations look OK to me, you have a typo u = (-1,4,7) not (-1,3,7). verifying (f1,f2,f3) is orthogonal is just a matter of computing the 9 inner products, or, you could notice that the change of basis matrix U has the property that U^-1 = U^T, and is thus orthogonal, and preseves inner products.
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  3. #3
    Junior Member
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    oops missed the typo, guess i need a break ... thanks for the help!
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