Linear Algebra: Eigenvectors and Orthonormal Bases

[Niles]

Homework Statement

Consider a symmetric (and hence diagonalizable) n x n matrix A. The eigenvectors of A are all linearly independent, 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 independent, will they also be orthogornal?

 

[CompuChip]

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.

 

[Niles]

1) They ar elinearly dependent, so yes - I guess I answered my own question there!

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

 

[CompuChip]

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).