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Final exam: Math 232
December 11th, 8:30am–12:30pm
Instructions/Remarks:
- Read all instructions.
- The questions are on the 12 numbered single-sided pages following this one. Count them now.
- Please include all details and rough work. You may use the back of the question pages and there
is extra paper should you require it.
- The exam is out of a total of 100 marks. The value of each question is indicated below.
- No calculators or electronic devices of any kind are permitted.
- You have 3 hours to complete this examination.
- Good luck!
Marks:
Total: /100 Grade:
Name: SFU e-mail ID:
Student Number:
Signature:
Question 1 [15pts] Suppose the entire population of a certain nation lives either in the city or the
country. Consider the following model for how the population in the city and country changes year
by year.
- Every year 10% of the people living in the city move to the country and the rest remain in the
city.
- Every year 20% of the people living in the country move to the city and the rest remain in the
country.
Let x
(k) 1 be the number of people living in the city in year^ k^ and let^ x
(k) 2 be the number of people
living in the country in year k. Let x
(k)
[
x
(k) 1
x
(k) 2
]
a. The model can be expressed as x
(k+1) = Ax
(k)
. What is A?
b. One of the eigenvalues of A is 1. What is the other eigenvalue?
c. Give an eigenvector corresponding to each eigenvalue of A.
d. Give an invertible matrix P and a diagonal matrix D, both of size 2 × 2 such that AP = P D.
Question 2 [15 pts] Let T : R
2 → R
2 be the linear transformation that reflects each point through
the line x = y and then stretches it in the x direction by a factor of 2.
a. What is the standard matrix for T?
b. What is the matrix for T relative to the basis B =
{[
]
[
]}
c. State a basis C for which the matrix for T relative to the basis C is diagonal.
(Continue work here)
Question 4 [5 pts] Let a, b, c, d, e, f, g, h, i be real numbers such that
a b c
d e f
g h i
State the following determinants. You do not need to justify your answer.
a.
d e f
g h i
a b c
b.
a b c
3 d 3 e 3 f
g h i
c.
a b 0
3 d 3 e 0
g h 0
d.
a − 2 d b − 2 e c − 2 f
d e f
g h i
e.
5 a 5 b 5 c
d e f
g − a h − b i − c
Question 5 [5 pts] Let B = {b 1 , b 2 , b 3 } be a basis for R
3
. Let T be a linear transformation on R
3
such that
T (b 1 ) = b 2 − b 3 , T (b 2 ) = b 1 − b 3 , T (b 3 ) = b 1 + b 2.
a. What is [T ]B, the matrix for T relative to the basis B?
b. Is there a vector x such that T (x) = 0?
Question 7 [10 pts] In each case, state whether the matrix A is invertible or non-invertible and briefly
justify your answer.
a. A =
b. A =
[
]
c. A is a diagonalizable 4 × 4 matrix and has eigenvalues 1, 1 , −i, i.
d. A is 3 × 3 and A
T
e. A is n × n and Col A = R
n .
Question 8 [15 pts] Consider the matrix
A =
a. Compute an orthonormal basis for Col A.
b. Let bˆ be the orthogonal projection of the vector b =
onto Col A. Compute bˆ.
c. Find a least-squares solution of system Ax = b.
Question 9 [15 pts] Suppose A has eigenvalues λ = 0.9 and λ = 1.1 with corresponding eigenvectors
v 1 =
[
]
and v 2 =
[
]
a. Give diagonal D and invertible P such that A = P DP
− 1 .
b. Find a formula for A^1000. (Your answer should be a 2 × 2 matrix with entries of the form
C 0 a
1000
1000 where C 0 , C 1 , a, b are constants.)
c. Let x
(k+1) = Ax
(k) where x
(0)
[
]
. For large k there is an approximate formula for x
(k) :
x
(k) ≈ γ
k
[
c
d
]
where γ, c, d are constants. State γ, c, d.
