Problem 4:

1 _ Problem 4 (30 points) Consider matrix A = [ ] with a > 0. (a) For what value of a does the null space N (A) become non-empty? Sketch N (A) together with the column space and the row space. d (b) When N (A) is non-empty, solve d: = Au for u(t) = [:8 E R2 under initial . . 36(0) [3] condltlon u 0 = = . ( ) [32(0) 3 (c) If we change the initial condition and allow u(0) to be anywhere on a circle x2 +y2 = r2, where should u(0) be so that \"(0), u(oo), and the origin are collinear? Express your result in terms of the slope y(0)/x(0)