(b) Suppose A is a real matrix and the vector is an cigenvector of A corresponding to the eigenvalue 2 + 8 What is the responser(k) of the system to the initial condition r(0) -1 ? 0 Exercise 6 Consider a 5 x 5 matrix where u and w are 5 x 1 matrices. (a) What is the rank off? (b) What is the nullity off? (c) What are the cigenvalues off if o'w negationslash Exercise 7 Recall the 2 pendulum cart system. Consider the equilibrium configurations defined by E1 : (if , 6, 5) = (0 0, 0) E2 : Consider state space representations of the linearizations corresponding to the following combinations of parameters and equilibrium conditions: L1 P1 E1 L2 P1 E2 L3 P2 E1 L4 P2 L5 P3 E1 LG P3 L7 P4 E1 L8 P4 E2 (a) Determine the eigenvalues of all the lincarized systems L1-L8. (b) Compare the behavior of the nonlinear system with that of the linearized system for cases L7 and L8. Illustrate your results with time histories of 0 and 02