# Orthonormal basis

• Apr 3rd 2011, 05:18 AM
mechaniac
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:

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

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

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 :)
• Apr 3rd 2011, 06:05 AM
Deveno
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.
• Apr 3rd 2011, 06:30 AM
mechaniac
oops missed the typo, guess i need a break :D ... thanks for the help!