# Linear algebra: Vector spaces

Niles

## 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

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.

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

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