课程名称︰线性代数一 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）︰2015/11/11 考试时限（分钟）：130 试题 : Notation: F denotes the set of complex numbers, n is a positive integer and I n is the identity matrix in M (F). n Part A 2 3 Problem 1 (10 pts). Let T: F → F be the linear transformation such that T((1, 2)) = (1, 1, 1); T((2, 1)) = (1, -1, 3). Let A = {(-5, 2), (-1, 1)} and B = {(2, 3, 1), (1, 2, 1), (1, 1, 1)} be bases of 2 3 F and F . Find the matrix representation [T] of T with respect to A and B. A, B 4 3 Problem 2 (20 pts). Let T: F → F be the linear transformation defined by T(v) = A・v, where ╭ 3 -1 1 1 ╮ │ │ A = │ 1 1 -1 3 │ ∈ M (F). │ │ 3 ×4 ╰ 2 0 0 2 ╯ (1) Find the rank and the nullity of T. (2) Find bases of Ker T and Im T. 3 Problem 3 (10 pts). Let V be the subspace of F spanned by a (1, 2, a), (1, a, 2), (a, 1, 2). Determine all possible values a ∈ F such that dim V = 2. F a Problem 4 (15 pts). Let ╭ -1 2 -2 ╮ │ │ A = │ 7 -3 1 │. │ │ ╰ 11 -8 6 ╯ -1 Find an invertible matrix P ∈ M (F) such that P AP is a diagonal matrix. 3 Problem 5 (10 pts). Let ╭ 1 0 1 ╮ ╭ 1 1 1 ╮ │ │ │ │ A = │ -1 -3 1 │; B = │ 1 2 1 │. │ │ │ │ ╰ 1 4 1 ╯ ╰ 1 4 3 ╯ Find all roots of the characteristic polynomial of AB. Part B Problem 6 (10 pts). Let V be a finite dimensional vector space over F with dim V = m. Let v , v , ..., v be n vectors in V. Show that if n ≧ m + 2, then F 1 2 n n there exists α , α , ..., α in F not all equal to zero such that Σ α v = 1 2 n i=1 i i n 0 and Σ α = 0. i=1 i Problem 7 (10 pts). Let A ∈ M (F). If rank A + rank(I + A) = n, show that n n Tr(A) + rank A = 0. Problem 8 (15 pts). Let A, B ∈ M (F). Let C = AB - BA. If CA = AC, prove that C n is NOT invertible. -- 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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