Recall that two vectors v and w are linearly independent if the only solution to the vector equation av + bw = 0
is the solution a = b = 0.
Let v and w be eigenvectors of a matrix M with corresponding nonzero eigenvalues λ and µ such that λ ≠ µ.
(i) Given scalars a and b, ﬁnd an expression for, M(av + bw), that does not contain M.
(ii) Hence or otherwise, prove that the eigenvectors v and w must be linearly independent.