Show from first principles, Le., by using the definition of linear inde- pendence, that if µ = z + iy, y # 0 is an eigenvalue of a real matrix A with associated eigenvector e=u+ iw, then the two real solutions Y() = "(u cos dt – w sin be) and Z(1) = e“(usin de + w cos be) are linearly independent solutions of X = AX. (b) Use (a) to solve the system x - (; )x. NB: Real solutions are required. Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate parts, then any one of the two eigenvalues yields two linearly indepen- dent solutions. The second will yleld 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.

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.