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Linear Algebra hw help

  1. Nov 6, 2007 #1
    1. The problem statement, all variables and given/known data
    Prove that any basis of R^n is an orthonormal basis with respect to some inner product. Is the inner product uniquely determined?


    2. Relevant equations
    I am not sure where to begin. Should I just define an arbitrary basis for a arbitrary R^n? I mean I think I understand the question about the inner product being uniquely determined but I am not sure where to begin.


    3. The attempt at a solution
    See above.
     
  2. jcsd
  3. Nov 6, 2007 #2
    How do you define inner products in R^n? A familiar question: how and when do symmetric matrices induce inner products on R^n?

    Uniquely determined means that there is no other inner product that has those properties. That is, given a basis there is only one inner product that makes the basis an orthonormal set. If you don't know what I meant by symmetric matrices you can just play around with scaling inner products by positive reals.
     
    Last edited: Nov 6, 2007
  4. Nov 6, 2007 #3
    Do you mean in (x^T)Kx or in notation <x,x> or in formula? I'm not going to lie I'm a bit confused with what you're asking.
     
  5. Nov 6, 2007 #4
    Do you mean by bilinearity, symmetry, and positivity?
     
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