课程名称︰线性代数二 课程性质︰数学系大一必修 课程教师︰谢铭伦 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2016/06/15 考试时限（分钟）：140 试题 : Part A t Problem 1 (20 pts). Find an orthogonal matrix P such that P AP is diagonal, where ╭ 4 2 2 ╮ │ │ A = │ 2 4 2 │. │ │ ╰ 2 2 4 ╯ Problem 2 (20 pts). Let ╭-3 0 ╮ │ │ A = │ 3 -3 │ ∈ M (|R). │ │ 3×2 ╰ 0 3 ╯ (1) Find the singular value decomposition (S.V.D.) of A and the Moore-Penrose + inverse A of A. (2) Show that there is no solution to the following linear equation ╭ 1 ╮ │ │ 2 Ax = │ 2 │, x ∈ |R . │ │ ╰ 3 ╯ Find the best approximate solution to the above equation. Problem 3 (20 pts). Let ╭ 1/4 1/4 1/2 ╮ ╭ 1 ╮ │ │ │ │ A = │ 1/4 1/4 1/4 │ and w = │ 1 │. │ │ │ │ ╰ 1/2 1/2 1/4 ╯ ╰ 1 ╯ (1) Find the Perron-Frobenius vector v of A. A n (2) Find lim A w. n→∞ Problem 4 (25 pts). Let V = M (|R) be a four-dimensional vector space over |R. 2 ╭ 1 2 ╮ ╭ 1 0 ╮ Let A = │ │ and B = │ │. Define the linear transformation L: V → ╰ 0 1 ╯ ╰ 3 1 ╯ V by L(X) = AXB. (1) (10 pts) Choose your favorite basis \mathscr{A} of V. Write down the matrix [L] ∈ M (|R) for L with respect to this basis \mathscr{A}. \mathscr{A} 4 (2) (10 pts) Determine the Jordan canonical form of L. (3) (5 pts) Let Q: V → |R be the quadratic form defined by Q(X) := det X. Determine the signature of Q. Problem 5 (15 pts). Let T: V → V be an isometry for the quadratic space (V, Q) in the previous problem. Show that there exist invertible matrices C, D ∈ t M (|R) such that either T(X) = CXD or T(X) = CX D. 2 Part B (Caution: No partial credit for bonus problems) Problem 6 (10 pts). Let A, B, C ∈ M (|R). If ABC = 0 and rank B = 1, show that n n either AB = 0 or BC = 0 . n n p+1 Problem 7 (10 pts). Let A ∈ M (|R). Suppose that A = A, where p is a prime n number. Show that 2 p-1 rank(A - I ) = rank(A - I ) = ... = rank(A - I ). n n n Problem 8 (10 pts). Is it possible to find polynomials a (x), a (x), a (x) ∈ 1 2 3 |C[x] and b (y), b (y), b (y) ∈ |C[y] such that 1 2 3 2 2 3 3 1 - x - y + 2xy + x y + x y = a (x) b (y) + a (x) b (y) + a (x) b (y)? 1 1 2 2 3 3 // 打不出来的字用LaTeX code呈现 -- 移居二次元(|R^2)的注意事项： 3. 如果你在从事random walk，往上下左右的 1. connectedness不保证pathwise connec- 的机率都是1/4，则你能回家的机率是1。 tedness。可能你跟你的幼驯染住很近， 4. 下面这个PDE是二次元上的波方程式 却永远没办法到她家。 http://i.imgur.com/2H9HllP.png 2. ODE的C^1 autonomous system不会出现 它的解不满足Huygens' principle，因此 chaos，在预测事情上比较方便。 讲话时会听到自己的回音，很不方便。 --

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