MIDTERM
EXAMINATION
Math-152 Winter- 2007/08
March 13 2008
University Of British Columbia
Name: __________________________________________________________
ID Number:______________________________________________________
Instructor: _______________________________________________________
Instructions
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You should have six pages including this cover.
There are 5 questions numbered (1)-(5).
Use this booklet to answer questions.
Return this exam with your answers.
Please show your work. Correct intermediate steps may earn credit.
No calculators are permitted on the test.
No notes are permitted on the test.
Maximum score= 25 Marks (attempt all questions)
Maximum Time= 50 minutes.
B E ST
OF
LUCK
1.
2.
3.
4.
5.
Total
5
5
5
5
5
25
1
MATH 152. Midterm 2.
2
Question 1.
(a) [2 points] Find the rank of the following matrices A and B. If needed,
perform row operations.


1 0 0 0 0
A= 0 1 1 0 0 
0 0 0 1 2


0 0 0
B= 1 0 0 
0 1 0
(b) [1 point] Find a nonzero solution for the homogeneous equations, A~x =
~0, and B~y = ~0.
µ
(c) [2 points] If
5 −2 5
3 8 8
¶
µ
~y =

2
1
¶
, then find the value of:


5 3 µ ¶
9 
~y ·  −2 8 
5
5 8
MATH 152. Midterm 2.
3
Question 2. [5 points] The weather in Vancouver is either good, indifferent, or bad
on any given day. If the weather is good on any day, then on the next
day, there is a 60% chance the weather will be good, a 30% chance
the weather will be indifferent, and a 10% chance the weather will be
bad. If the weather is indifferent on any day, then on the next day,
it will be good with probability 0.40, indifferent with probability 0.30,
and otherwise bad. Finally, if the weather is bad on any day, then
on the next day it will be good with probability 0.40, indifferent with
probability 0.50, and otherwise bad. Suppose that on Monday, there is
a 50% chance of good weather and a 50% chance of indifferent weather.
What are the chances for good weather on Wednesday?
MATH 152. Midterm 2.
4

1
 1
Question 3. [5 points] Let B = 
 1
0
3
be any vector in R .


1 1
0
 −4
0 1 
, ~c = 
 0
1 0 
2 −1
1


, and ~x is allowed to

(a) Find the vector ~x such that kB~x − ~ck is minimal.
(b) What is the minimal possible distance kB~x − ~ck?
MATH 152. Midterm 2.
5
Question 4. [5 points] Suppose that the transformation T acts on each vector ~x of
R2 as follows: first ~x is rotated counterclockwise by 90 deg, and then
the resulting vector is projected onto the line y = 3x.
(a) Find the matrix representing T .
(b) Determine whether this matrix is a projection matrix, Projα , for some
angle α. Explain why, or why not.
(c) Compute T (T (~x)).
MATH 152. Midterm 2.
6


−2 1 p
Question 5. [5 points] The matrix B =  0 −1 1  contains a parameter p.
1
2 0
(a) For which values of the parameter p is B invertible?
(b) For those values of p for which the inverse exists, find a formula for
B −1 in terms of p.