课程名称︰工程数学-线性代数 课程性质︰必修 课程教师︰冯世迈 开课学院：电机资讯学院 开课系所︰电机系 考试日期（年月日）︰2018/03/28 考试时限（分钟）：50分钟（09:20-10:10）-最後助教多给5分钟检查 试题：用右上角的*，表示向量(即课本/上课的粗体) 1. Consider Ax*=b*, where the augmented matrix is given by – – | 1 -1 2 2 2 | | -1 1 -2 2 6 | [ A b*]=[ a1* a2* a3* a4* a5* b*]=| 1 -2 2 0 1 | | -2 1 -4 2 11 | – – (a) (20%) Find the reduced row echelon form of [ A b* ]. (b) (10%) Find the general solution of Ax*=b* in vector form. (c) (4%) What are the rank and nullity of [ A b* ] ? (d) (6%) Choose 3 column vectors from A to form a new 4x3 matrix A' so that A'x* = 0* has only zero solution. (e) (10%) Explain why Ax* = -2b*-6.58a2* is consistent. Find its general solution in vector form. 2. (20%) Find the inverse of the following matrix. – – | 1 2 1 | | 2 5 1 | | 2 4 1 | – – 3. Let A = [ a1* a2* a3* a4* a5*] with the following reduced row echelon form: – – | 1 -1 2 2 2 | | -1 1 -2 2 6 | R = | 1 -2 2 0 1 | | -2 1 -4 2 11 | – – (a) (5%) Find the reduced row echelon form of A' = [a2* a1* a3* a4* a5*] (b) (5%) Find the reduced row echelon form of [ I4* A ] (c) (10%) If a1*=[1 1 1 1]^T, a3*=[1 2 0 1]^T, a4*=[0 1 1 1]^T, find A. 4. (10%) Let A and B be m x n and n x p matrices respectively. Prove that the span of the columns of AB is a subset of the span of the columns of A. --

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