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Homework Help: Linear algebra: Vector spaces

  1. Aug 17, 2008 #1
    1. The problem statement, all variables and given/known data
    Consider a symmetric (and hence diagonalizable) n x n matrix A. The eigenvectors of A are all linearly independant, and hence they span the eigenspace Rn.

    Since the matrix A is symmetric, there exists an orthonormal basis consisting of eigenvectors.

    My questions are:

    1) Will this orthonormal basis of eigenvectors also span the same space Rn?

    2) If two vectors are linearly independant, will they also be orthogornal?
     
  2. jcsd
  3. Aug 17, 2008 #2

    CompuChip

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    For 1): when does a set of vectors span the vector space? Do the eigenvectors satisfy these conditions? [Actually, you already gave the answer yourself... do you see where? ]

    For 2): Consider (1, 0) and (1, 1) in R2.
     
  4. Aug 19, 2008 #3
    1) They ar elinearly dependant, so yes - I guess I answered my own question there!

    2) Great, a counter-example, so no. Thanks!
     
  5. Aug 19, 2008 #4

    CompuChip

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    1) Yep, it follows from the fact that "the eigenvectors are linearly independent" and that there are n of them. That is, they form a basis, as you said in the question. And of course a basis always spans the space (even a non-orthogonal and/or not normalized one).
     
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