text: 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!
explanation: The document contains a math question and an answer. The question involves showing that a given set of vectors forms an orthonormal basis and determining the coordinates of a vector in this new basis. The user also provides their attempt at solving the problem and receives feedback from another user.
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distilabel_metadata.role: system
distilabel_metadata.content: 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!
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and an answer. The question involves showing that a given set of vectors forms an orthonormal basis and determining the coordinates of a vector in this new basis. The user also provides their attempt at solving the problem and receives feedback from another user.
  
  Label: question-and-answer