1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor
    Homework Helper

    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).
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Linear algebra: Vector spaces