课程名称︰线性代数二 课程性质︰必修 课程教师︰林绍雄 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2012/6/16 考试时限（分钟）：180 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : There are problems A to F with a total of 140 points. Please write down your computational or proof steps clearly on the answer sheets. ----------------------------------------------------------------------------- A. Consider A = ┌ 1 -a ┐ , where a >=0 is a constant. └ -4a 1 ┘ (a) (10 points) Write down the associated Jacobi matrix J and the SOR-matrix Lω. Compute their eigenvalues, and find ρ(J) and p(Lω). (b) (10 points) If 0 < ω < 1, show that the Jacobi method converges iff the SOR-method converges iff A is a M-matrix. (c) (5 points) Find the optimal parameter ωopt for the SOR-method. ----------------------------------------------------------------------------- B. (15 points) A feed-mix company is preparing a mixture of three feeds, feed 1, feed 2 and feed 3. Each unit of feed 1 contains 1 gram of protein, 2 grams of fat, and cost 20 cents; each unit of feed 2 contains 2 grams of protein, 2 grams of fat, and cost 30 cents; each unit of feed 3 contains 2 grams of protein, 1 gram of fat, and cost 25 cents. If the mixture of these three feeds must contain at least 200 grams of protein, and at least 150 grams of fat, how many units of each feed should the company use so as to minimize costs? ----------------------------------------------------------------------------- C. Consider the linear programming problem of minimizing x1 + x2 - x3 subjected to 2x1 - 4x2 + x3 <= 4, 3x1 + 5x2 + x3 <= 2 and xj >= 0 (j = 1, 2, 3). (a) (4 points) Apply whatever method (Farka's lemma, or the phase I method, or...) to show that this problem is feasible. (b) (9 points) Apply the simplex method to find an optimal feasible solution. (c) (7 points) Write down the dual problem. Apply the complementary slackness conditions to find the optimal feasible solution of the dual problem complemantary to the solution in (b). ----------------------------------------------------------------------------- D. (10 points) Find a Nash equilibrium for the two-person game with payoff matrix ┌ (3,1) (1,3) ┐. │ (2,2) (2,1) │ └ (1,1) (3,0) ┘ ----------------------------------------------------------------------------- E. (15 points) Players A and B play the scissor-paper-stone-glass-water game. The two players simultaneously choose one of the five objects. (有给图但不好画，以说明代替如下: 石头赢水，水赢杯子.剪刀.纸， 剪刀赢石头.杯子，杯子赢石头.布，布赢石头.剪刀) The winner gets $1 pay from the loser. Find the optimal mixed strategy for both players. ----------------------------------------------------------------------------- F. Work out the following problems. (a) (12 points) Let A ∈ M(n,F) be an irreducible Hessenberg matrix, and α∈ F. Perfrom the QR-decomposition A - αI = QR, and n define A' = RQ + αI . If α 不属於 ρ(A), prove that A' n is also an irreducible Hessenberg matrix. (b) (8 points) The Jacobi method applied to a positive definite matrix always converges. Is this a true statement? (c) (10 points) Assume that A ∈ M(n,F) has a decomposition A = S - T with S, T ∈ M(n,F) such that S is invertible. If the sequence (k) (k+1) (k) x defined by the iteration scheme Sx = Tx + b (0) (k = 0, 1, ...) converges for all b, x ∈ F(n). Prove that A is nonsingular. T T (d) (15 points) Let A ∈ M(m ×n,R) have row vectors v1, ..., vm T T m (so that A = [v1...vm] ), and b = [b1...bm] ∈ R . n Define P = {x ∈ R | x >= 0, and Ax >= b}. Prove that x* ∈ P is an extreme point of P iff the set T {vi | i = 1, ..., m, vi x* = bi} contains n linearly independent vectors. (e) (10 points) There exists a nonzero payoff matrix A ∈ M(3 ×2,R) for a two-person zero-sum game whose Von Newmann value is 0, while the optimal strategies x*, y* for both players satisfy x* > 0, y* > 0. Is this a true statement? -- --

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