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(MATH111)[2010](s)midterm~cs_zxxab^_10440.pdf
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HKUST
MATH111 Linear Algebra
Midterm Examination
25th March 2010
Name:________________
Student I.D.:________________
9:00-10:20am
Directions:
. DO NOT open the exam until instructed to do so.

. Please write your Name and ID number in the space provided above.

. You may use a HKEA approved calculator, but graphical calculators are NOT allowed.

. All mobile phones and electronic equipments other than calculators should be switched off during the examination.

. This is a closed book examination.

. When instructed to open the exam, please check that in addition to the cover page, you have 4 questions on 4 pages, and several blank pages.

. Answer all questions.

. Show the working steps of your answers for full credit.

. Cheating is a serious offense. Student caught cheating are subject to a zero score as well as additional penalties.





Question No.
Points
Out of

Q. 1

7

Q. 2

8

Q. 3

7

Q. 4

8

Total Points

30




1. Let
, ........... 654321 A
....... 1-21-201 B

Compute (1) , (2) , (3) . AB
BA
TA

Solution:
(1)
.................................................................................. 4121-281-041- 162526051615142324031-413122122011-211 1-21-201 654321 AB

(2)
.............................................................. 001411 614221513211-624021523011 654321 1-21-201 BA

(3)
....... 642531 TA

2. Consider the linear system
. .......................2793327313205 4321432143214321xxxxxxxxxxxxxxxx

(a) Write down the matrix equation form of the linear system. bx.A

(b) Find all the solutions of the linear system.
(c) For the matrix in (a), does the equation have at least one solution for each c in ? Prove your answer. A
cx.A
4R

Solution:
(a)
............. 79-31271-332-111-51-1 A
............. x4321xxxx
............. 2310 b

(b)
02-213-27 00001-1002301025001 2310 79-31271-332-111-51-1 ..............................

Solutions of the linear system
..................444342412232132527xxxxxxxx
.......................................... 0221327 112325 44321xxxxx

(c) is matrix and has pivot positions. By the theorem, the equation does not always have at least one solution for any in . A
44.
43.
cx.A
c
4R

3. (a) Compute the determinant of the matrix:
. ............. 103-321-2102-42-1-32-21 A

(b) Do the columns of the matrix form a linear independent set? Prove your answer. A

Solution:
(a)
....31332 3112 11 3301-21120 411-1-21120 1 411-01-210120032-21 103-321-2102-42-1-32-21 12.................Adet

(b) Since , is invertible. By the theorem, the columns of the matrix form a linear independent set. 03...Adet
A
A

4. Let be a linear transformation such that 33RR.:T

. .................,,,,,321321321321313232323132323231xxxxxxxxxxxxT