Linear algebra: Vector spaces

  • Thread starter Niles
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  • #1
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Homework Statement


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?
 

Answers and Replies

  • #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.
 
  • #3
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1) They ar elinearly dependant, so yes - I guess I answered my own question there!

2) Great, a counter-example, so no. Thanks!
 
  • #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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