Do Real Symmetric Matrices have 'n' linearly independent eigenvectors? I know that Real Symmetric Matrices have real eigenvalues and the vectors corresponding to each distinct eigenvalue is orthogonal from this answer. But what if the matrix has repeated eigenvalues? Does it have linearly independent (and orthogonal) eigenvectors? How to prove that?
PS: In the answer I referred to has another answer which might have answered this question. I'm not sure if it answered my question since I didn't understand it. If it did answer my question, can anyone please explain it?
Thanks!

Real Symmetric Matrices have $n$ linearly independent and orthogonal eigenvectors. 
There are two parts here. 
1. The eigenvectors corresponding to distinct eigenvalues are orthogonal which is proved here. 
2. If some of the eigenvalues are repeated, since the matrix is Real Symmetric, there will exist so many independent eigenvectors. (Proof here and here.) As John Ma pointed out, in this case we can use Gram–Schmidt orthogonalization to get orthogonal vectors. 
So, all $n$ eigenvectors of a Real Symmetric matrix are linearly independent and orthogonal
