Linear Algebra

• 5. (opts) Consider the subspace W = {p(x) E P3 I p(2) = 0 and p'(2) = 0} of P3. Find a basis for W. 6. (6pts) The 5 x 5 matrix A has two eigenvalues, Ai = 1.0 and Al = —0.7. An eigenvector corresponding to i1 11 2 12 Ai is 171 = 3 and an eigenvector corresponding to A2 is 62 = 13 . Suppose 1 = 6171 + 4-112- 4 14 5 15 (a) Find A3E. (b) Find lim Aki. k—■co 7. (9pts) A linear model for the population of cheetahs and gazelles in Namibia is given by the following pair of equations: 0.3 Ck + 0.4 Ck = Ck+1 —p Ck 1-3 Ck = Ck+1 $ where Ck measures the number of cheetahs present in a certain Namibian game reserve at time k, Gk gives the number of gazelles (measured in tens), and k is measured in months. (a) Find a value for p (to three decimal places) that guarantees a steady-state outcome for this model, then determine the number of cheetahs present for every 1000 gazelles in the long-run. (b) Find a value for p (to three decimal places) that will guarantee 2% growth in both populations. (c) What must be true about the eigenvalues of the transition matrix if both populations die out in the