Suppose for a given matrix A, Sarah finds the eigenvectors v1 = [1 3 4 5]' and v2 = [5 6 3 4]' form a base for eigenspace of labmda = 2. Now suppose Janie finds the eigenvectors v3 = [1 2 2 3]' and v4 = [7 8 7 6]' form a base for eigenspace of lambda = 4. Is Janie's solution compatible with Sarah's? Okay, so I know that if v1 and v2 form a base for the eigenspace of lambda, they must be linearly independent. This same fact goes for v3 and v4. Now, my question is, to check whether or not Janie's solution is compatible with Sarah's, would I simply make sure that they are all linearly independent? If so, then they are compatible? The part that's throwing me off is that they belong to different eigenvalues. So, can you have the same vector (or a linear combination of it), only belonging to a different eigenvalue of the same matrix? Or are the vectors unique to the eigenvalues? Help much appreciated.