# Difference between revisions of "Science:Math Exam Resources/Courses/MATH152/April 2015/Question B 2 (c)/Solution 1"

Suppose that  are eigenvectors corresponding to distinct eigenvalues , respectively. Let  be the largest integer for which  are linearly independent. If we had , we could write  for some  not all zero. Now , whence . As the  are distinct, this implies (by linear independence) that , so , contradicting the assumption that  was an eigenvector. We conclude that , so all the eigenvectors are linearly independent.
Each of the 3 eigenvalues found in part (b) has a corresponding eigenvector, and by the above, they are linearly independent. (Clearly, there cannot be more linearly independent eigenvectors, since the dimension of the matrix is 3.) Thus, the answer is