We know that a normal matrix will have, or can be made to have (by orthogonalization, if not all its eigenvalues are distinct), orthogonal eigenvectors. This means that they are also linearly independent (i.e. they form a basis for the vector space concerned). However, in the case when these eigenvectors are not all orthogonal (i.e. when some share the same, degenerate, eigenvalues and weren't orthogonal and we did not orthogonalize them), will they still be linearly independent?
In other words, are the eigenvectors of a normal matrix always linearly independent, regardless of their mutual orthogonality?