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