Orthonormal basis

• Apr 3rd 2011, 05:18 AM
mechaniac
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 :)
• 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!
The text passage contains a mathematical problem involving orthonormal bases and a solution to part of the problem. It asks to show that a given set of vectors forms an orthonormal basis and to determine the coordinates of a vector in this new basis. An answer to the second part is provided, but the first part is only mentioned as solved after the post.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Orthonormal basis

• Apr 3rd 2011, 05:18 AM
mechaniac
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 :)
• 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!
user
The text passage contains a mathematical problem involving orthonormal bases and a solution to part of the problem. It asks to show that a given set of vectors forms an orthonormal basis and to determine the coordinates of a vector in this new basis. An answer to the second part is provided, but the first part is only mentioned as solved after the post.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS