Question 5: 20 Marks (5.1) Show from first principles, i.e., by using the definition of linear inde- pendence, that if u = r + iy, y * 0 is an eigenvalue of a real matrix A with associated eigenvector v = u + iw, then the two real solutions Y(t) = eat(ucosbt - w sin bt) and Z(t) = eat (usin bt + w cos bt) are linearly independent solutions of X = AX. 2 (5.2) Use (a) to solve the system * = (8 4) x. NB: Real solutions are required. Hint: Recall that if the roots of C(X) = 0 occur in complex conjugate parts, then any one of the two eigenvalues yields two linearly indepen dent solutions. The second will yield solutions which are identical (up to a constant) to the solutions already found. Check Theorem 2.19 (Equation 2.2) and Example 2.20 on page 32 of the study guide in this regard