课程名称︰工程数学-线性代数 课程性质︰统一教学 课程教师︰冯世迈 开课学院：电机资讯学院 开课系所︰电机工程学系 考试日期（年月日）︰2009.03.19 考试时限（分钟）：50 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Use of all automatic computing machines including calculator is prohibited 1. Let the matrix A and the vector b be respectively defined by [ 1 2 -1 2 0 1] [2] A = [a1 a2 a3 a4 a5 a6] = [-1 -2 1 2 0 3] , b = [6] [ 2 4 -3 2 0 0] [3] (a) (20%) Find the reduced row echelon form [R c] of [A b]. (b) (10%) What are the rank and nullity of A? (c) (10%) Is the set {a1, a3, a6} linearly independent? Explain your answer. (d) (10%) Is the equation Ax = b consistent? If it is, find its the general solution in vector form. (e) (10%) From the reduced row echelon form [R c], determine if the vector b is in the span of {a1, a3, a4} or not? If it is, write b as a linear combination of a1, a3, a4. (f) (10%) From the reduced row echelon form [R c], can you find the reduced row echelon form of the 3 × 4 matrix A' = [a1 a4 a3 a6]? 2. (20%) Let A = [a1 a2 a3 a4 a5 a6] be the 3 × 6 matrix defined in Problem 1. Choose 3 vectors from {a1, a2, a3, a4, a5, a6} to form an invertible 3 × 3 matrix. Find its inverse. 3. (10%) Suppose that B and C are 3 × 3 matrices, and B is invertible. Find the reduced row echelon form of (a) [B I3]; and (b) [C] [B] (Express your answer in terms of O, I3, B, B^(-1), and C). --

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