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）-最后助教多给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.