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?