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How to count all the orthogonal transformations?

  1. Jun 1, 2013 #1
    1. The problem statement, all variables and given/known data
    Find an orthogonal transformation ##\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}## that map plane ##x+y+z=0## into ##x-y-2z=0## and vector ##v_{1}=(1,-1,0)## into ##(1,1,0)##. Count all of them!

    2. Relevant equations
    ##A_{S}=PA_{0}B^{-1}##


    3. The attempt at a solution
    So basis ##B=\begin{bmatrix}
    1 & 1 & 1\\
    1& -1 & 1\\
    1& 0& 2
    \end{bmatrix}## and ##P=\begin{bmatrix}
    1 & 1 & 1\\
    -1& 1 & -1\\
    -2& 0& 1
    \end{bmatrix}## where the last vector in both basis is a vector product of ##n_{1} \times v_{1}##

    ##A_{0}=diag(1,1,1)## and ##A_{S}=PA_{0}B^{-1}##

    Now, how do I count them? What do they represent - meaning, how do they differ from this one? o_O please help.
     
  2. jcsd
  3. Jun 1, 2013 #2

    tiny-tim

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  4. Jun 1, 2013 #3
    So whatever I do, ##det(A_{0})=1## has to stay the same, where ##A_{0}=diag(1,1,1)##. Meanin I could also use ##A_{0}=diag(-1,-1,1)## or ##A_{0}=diag(-1,1,-1)## or ##A_{0}=diag(1,-1,-1)##, right?

    To sum up, there are 4 orthogonal transformations that do that?
     
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