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Linear algebra - Orthogonal matrix

  1. Apr 14, 2010 #1
    1. The problem statement, all variables and given/known data
    Let T: Rn -> Rn be a linear transformation, and let B be an orthonormal basis for R^n. Prove that [ the length of T(x) ] = [ the length of x ] if and only if [T]B (the B-matrix for T) is an orthogonal matrix.


    2. Relevant equations
    None I don't think.


    3. The attempt at a solution
    I've always had a hard time grasping linear transformation problems, so I don't even really know where to start with this one.
     
  2. jcsd
  3. Apr 15, 2010 #2
    A transformation is orthogonal if its matrix representation obeys a certain equation. In an orthonormal basis, this equation reduces to T-1=TT. (It's inverse is equal to its transpose.) So that's one relevant equation. How would you express, as a matrix equation, the fact that the length of a vector is unchanged by a transformation? Hint: if the length stays the same, what can we say about the square of the length?
     
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