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）︰2016/04/20 考试时限（分钟）：110 试题 : Part A Problem 1 (10 pts). Let 2 3 V = {a + a x + a x + a x | a , a , a , a ∈ |R} 0 1 2 3 0 1 2 3 be the vector space of real polynomials of degree ≦ 3. Let 〈, 〉 be the inner product on V given by 1 〈f, g〉 = ∫ f(t)g(t)dt, f, g ∈ V. 0 Let W = |R．1 be the one-dimensional vector spanned by constant functions. Find ⊥ an orthonormal basis of W . 4 Problem 2 (10 pts). We equip |R with the standard inner product and let V be 4 the subspace of |R given by 4 V = {(x , x , x , x ) ∈ |R | x = x + x , x = x - x }. 1 2 3 4 1 3 4 2 3 4 Let v = (1, 2, -1, 1). Find the orthogonal projection of v onto V. Problem 3 (25 pts). Let ╭ 1 1 -1 ╮ A = │ │. ╰ 1 1 -1 ╯ (1) (15 pts) Find the singular value decomposition (S.V.D.) of A. (2) (5 pts) Find the Moore-Penrose inverse of A. (3) (5 pts) Let W = {(x , x , x )| x + x - x = 0} be a subspace of the real 1 2 3 1 2 3 3 vector space |R with the standard inner product space. Find the dimension and a ⊥ basis of W . Problem 4 (25 pts). Let ╭ 8 2 -2 ╮ │ │ A = │ 2 5 4 │. │ │ ╰-2 4 5 ╯ t t (1) (15 pts) Find P ∈ M (|R) such that P P = I and P AP is diagonal. 3 3 (2) (10 pts) Suppose that B ∈ M (|R) and C ∈ M (|R) such that BC = A. 3 ×2 3 ×2 Find CB. Part B Problem 5 (15 pts). Let (V, 〈, 〉) be a finite dimensional inner product space over €. Let T: V → V be a linear transformation. Show that T is self-adjoint if and only if for every v ∈ V, 〈T(v), v〉 is a real number. Problem 6 (15 pts). Let V, U and W be finite-dimensional inner product spaces over F. Let S: U → V and T: V → W be linear transformations such that im S = ker T. Show that SS* + T*T is an isomorphism from V to V. -- 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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