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Homework Help: Forming an orthogonal matrix whose 1st column is a given unit vector

  1. Nov 30, 2006 #1
    1. The problem statement, all variables and given/known data
    Show that if the vector [tex]\textbf{v}_1[/tex] is a unit vector (presumably in [tex]\Re^n[/tex]) then we can find an orthogonal matrix [tex]\textit{A}[/tex] that has as its first column the vector [tex]\textbf{v}_1[/tex].

    3. The attempt at a solution
    This seems to be trivially easy. Suppose we have a basis [tex]\beta[/tex] for [tex]\Re^n[/tex]. We may apply the Gram-Schmidt orthogonalization process to [tex]\beta[/tex] with [tex]\textbf{v}_1[/tex] as the generating vector and normalise the resultant orthogonal basis to obtain an orthonormal basis [tex]\gamma[/tex]. Choose [tex]\textit{A}[/tex] such that the elements of [tex]\gamma[/tex] comprise the columns of [tex]\textit{A}[/tex].

    QED

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    So what am I overlooking?
     
    Last edited: Nov 30, 2006
  2. jcsd
  3. Dec 1, 2006 #2

    HallsofIvy

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    Science Advisor

    Nothing- although I would say "Let [itex]\beta[/itex] be a basis for R containing v1..."
     
  4. Dec 1, 2006 #3

    StatusX

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    Homework Helper

    Another candidate is any rotation matrix which rotates the vector [1,0,0,...] into v_1.
     
  5. Dec 1, 2006 #4
    Thanks. I thought I had made a huge assumption somewhere.

    StatusX, yes that's a simpler possibility indeed.
     
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