(MATH2111)[2012](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 turn o. all phones and pagers, and remove headphones.

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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.

Please read the following statement and sign your signature.

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

governing academic integrity.

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

.

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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:

.

.

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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