1. ## 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

2. 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.

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