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(MATH2111)(f)midterm~=2qq1j^_80392.pdf
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HKUST
MATH2111 Introduction to Linear Algebra
Midterm Examination Name:
31st Oct 2012 Student ID:
18:00-19:00 Lecture Section:

Directions:
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DO NOT open the exam until instructed to do so.

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This is a closed book and notes examination.

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Please write your name, ID number, and lecture section in the space provided above.

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When instructed to open the exam, please check that you have 6 pages (excluding the cover page) of 4 questions.

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Answer all questions. Show an appropriate amount of work for each problem. If you do not show enough work, you will get only partial credit.

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You may write on the backside of the pages, but if you use the backside, clearly indicate that you have done so.

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You may use an ordinary scienti.c calculator approved by HKEAA, but calculators with graphical, symbolic algebra or calculus functions are NOT allowed.

I have neither given nor received any unauthorized
aid during this examination. The answers submitted
are my own work.

I understand that sanctions will be imposed, if I am
found to have violated the Universitys regulations

Students Signature :
Question No. Points Out of
Q. 1 8
Q. 2 12
Q. 3 10
Q. 4 10
Total Points 40

1. ([8 points]) Let AT be the transpose of the matrix A and let
11 1 0
A = .
10 .11
(a) Compute the matrix product AT A. [4 pts]
Solution
The product AT A is 4 4 matrix. And the product is
.
.
AT A =
...

2 1 0 1
1 1 1 0
0 1 2 .1
1 0 .1 1

...

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(b) Determine whether the columns of AT A are linearly independent or not. State your reason. [4 pts]
Solution
Let A =[a1, a2, a3, a4] and AT A =[AT a1,AT a2,AT a3,AT a4]. Since the columns of A are
linearly dependent, the columns of AT A are linearly dependent.
Another solution is to perform row operation on the matrix AT A and .nd two pivots. Then
its another way to deduce that the columns of AT A are linearly dependent.

2. ([12 points]) Consider the augmented matrix:
.
.
.

1 .1 2 1 2
3 0 1 2 4
1 2 .3 k 2l

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(a) Determine all values of k such that it represents a consistent linear system. Point out the basic variables and free variables for such k and l values. [5 pts]
Solution
.
..
..
.
1 .
0
12 .3 k

2l 03 .5 k . 1 2l . 2 000 k 2l
The linear system is consistent when k .0 and l can be any values. The linear system is
=
also consistent when k = 0 and l = 0.
When k = 0, the basic variables are x1 and x2. The free variables are x3,x4.
When k .The free variable is x3.

= 0, the basic variables are x1, x2 and x4.
(b) Solve the linear system and express the solution in parametric vector form when k = 1 and l = 1. [4 pts]
Solution
When k = 1 and l = 1, performing row