课程名称︰线性代数 课程性质︰选修 课程教师︰黄维信 开课学院：工学院 开课系所︰工程科学与海洋工程学系 考试日期（年月日）︰2009.4.24 考试时限（分钟）：120 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : ┌2 3 3┐ 1. Let A=│0 5 7│ └6 9 8┘ (a) Factor A into LU.(b) Find a basis for the row space of A. (c) Find a basis for the column space of A.(15%) 2. True or false, with reason if true and counterexample if false:(25%) (a) If columns 1 and 3 of B are the same, so are columns 1 and 3 of AB. (b) If rows 1 and 3 of B are the same, so are rows 1 and 3 of AB. 2 2 2 (c) (AB)=A B (d) If A and B are symmetric then AB is symmetric. (e) If A and B are invertible then AB is invertible. 3. What is the relation between the rank r and the dimension of A when the → → number of solutions to Ax = b is (10%) → (a) 0 or 1, depending on b. → (b) ∞, independing of b. → (c) 0 or ∞, depending on b. → (d) 1, regardless of b. 4 4. Find a basis for the following subspaces of R .(15%) (a) The vectors for which X = 2X 1 4 (b) The vectors for which X +X +X = 0 and X +X = 0. 1 2 3 4 3 (c) The subspace spanned by (1, 1, 1, 1),(1, 2, 3, 4), and (2, 3, 4, 5). → T → T 5. Apply the Gram-Schmidt process to a = [0 0 1] , b = [0 1 1] , → T c = [1 1 1] , and write the result in the form A = QR.(15%) 6. Find the quadratic curve which gives the best least-squares fit to the 5 curve y = x between x = 0 and x = 1.(20%) --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 218.166.155.228