Linear Algebra

5. (opts) Consider the subspace W = {p(x) E P3 I p(2) = 0 and p'(2) 6. (6pts) The 5 x 5 matrix A has two eigenvalues, Al = 1.0 and AI = 1 2 Ai is vi = 3 and an eigenvector corresponding to A2 is = 4 5 = 0} —0.7. 11 12 13 14 15 of P3. Find a basis for W. An eigenvector corresponding to . Suppose x = 6vt + 4172. Mr 4 (a) Find A3i. (b) Find Um Aki. k—oo 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 —pCk + 1.3Gk = where Ck measures the number of cheetahs present in a certain Namibian game reserve at time k, Ck 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 eigenvalue.s of the transition matrix if both populations die out in the