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Orthogonal matrices prove: T is orthogonal iff [T]_bb is an orthogonal matrix

  1. Feb 22, 2010 #1
    1. The problem statement, all variables and given/known data

    Let B = {v1, ..., vn} be an arbitrary orthonormal basis of Rn, prove T is orthogonal iff [tex][T]_{BB}[/tex] is an orthogonal matrix.

    Hint: If B is orhtogonal basis for Rn then, [tex]x.y = [x]_B . [y]_B[/tex]for all x, y in Rn.

    3. The attempt at a solution

    If [tex][T]_{BB}[/tex] is an orthogonal matrix then

    1) [tex] ||[T(x)]_B|| = ||[x]_B|| [/tex]

    2) [tex] [T(x)]_B . [T(y)]_B = [x]_B . [y]_B[/tex]

    and since B is orthonormal,

    [tex] ||[x]_B|| = ||x||[/tex]

    [tex][x]_B . [y]_B = x.y[/tex]


    That's all I've got so far.. is this even right? How do I tie it into T being orthogonal?
     
  2. jcsd
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