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Math Help - Orthonormal basis

  1. #1
    Junior Member
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    Orthonormal basis

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

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

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

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

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

    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:

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

    u=(-1,3,-7)

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

    Thanks!



    Edit: dont need help to show that (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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