
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 :)

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.

oops missed the typo, guess i need a break :D ... thanks for the help!