課程名稱︰工程數學-線性代數
課程性質︰必修
課程教師︰馮世邁
開課學院：電機資訊學院
開課系所︰電機系
考試日期（年月日）︰2018/03/28
考試時限（分鐘）：50分鐘（09:20-10:10）-最後助教多給5分鐘檢查
試題：用右上角的*，表示向量(即課本/上課的粗體)
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.