4. Suppose em = Aim where A = (: 3 )a,b,c1d e a maze} = (:33 ) Use the denition of linear independence to justi:l"'r,.r each of the nliowing. Recall that if n E R" = i]- then each component of u must equal ll. {a} Suppose the eigenvalues of A are distinct and real-valued. Show that the terms in the solution nit] are linearly independent. {h} Suppose there is only one eigenvalue of A Shaw that the terms in the solution unit} are linearly independent. (Hint: There are two easees here - one case when there are two linearly independent eigenyeotnrs asmeiated with that eigenvalue and the semnd case is when we cannot nd two linearly independent eigenvalues] {1:} Suppose the eigemralues of A are complex oonjugates. Shaw that the terms in the solution zit} are linearly independent. 5. Unnairuet an example of a second-order UDE that can he solved. using a power series hut not he solved explicitly using other methods in this class. Solve this ODE using a power series