MATH 1853
Problem Set #1
1.
Solve the following two equations using Gaussian elimination:
2 x1 3x2 11x3 5 x4 2
x x 5x 2 x 1
1 2
3
4
2
x
x
3
x
2
x
3
4 3
1 2
x1 x2 3x3 4 x4 3
and
4 x1 x2 x4 4
x 4x x 1
1
2
3
x
4
x
x
3
4 4
2
x1 x3 4 x4 10
2.
1 2 3 4

TUT/1853/MATH1853/6
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Tutorial 6
1. Suppose that X follows N (3, 9), find
(a) P (2 < X < 5), (b) P (X > 0), and (c) P (|X 3| > 6).
2. A machine produces tubes of length 1m. Assume the length of the

TUT/1853/MATH1853/5
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Tutorial 5
1. It is known that screws produced by a certain company will be defective with a
probability 0.01 independently of each other. The company sells the screws in
packa

AS/1853/MATH1853/2
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Assignment 2 Combinations and Probability
Due Date: November 13, 2015 (6:30pm)
Please submit your assignment to the assignment box (4/F, Run Run Shaw Building)
1. In order to en

TUT/1853/MATH1853/4
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Tutorial 4
1. A ball is in any one of n boxes and is in the i-th box with probability Pi . If the
ball is in box i, a search of that box will uncover it with probability i . Fi

AS/1853/MATH1853/3
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Assignment 3 Probability and Distributions
Due Date: November 20, 2015 (6:30pm)
Please submit your assignment to the assignment box (4/F, Run Run Shaw Building)
1. Consider a fu

AS/1853/MATH1853/1
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Assignment 1 (Complex Numbers)
Due Date: November 6, 2015 (Friday) (6:30pm)
Please submit your assignment to the assignment box (4/F, Run Run Shaw Building)
1. Let z = 1 + 3i. F

AS/1853/MATH1853/4
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Assignment 4 Normal Distribution and Statistical Inference
Due Date: November 30, 2015 (6:30pm) 7 questions
Please submit your assignment to the assignment box (4/F, Run Run Sha

TUT/1853/MATH1853/1
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Tutorial 1
1. Let A = cfw_1, 2 and B = cfw_1, 3, 5, 6, 7.
(a) What are |A| and |B|?
(b) Find A B.
(c) Find A B.
(d) Write down all the subsets of A.
2. Let A = cfw_x2 : x Z and

AS/1853/MATH1853/3
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Assignment 3 Probability Distributions
Due Date: 23 April 2015 (4pm)
Please submit your assignment to the assignment box (4/F, Run Run Shaw Building)
1. (a) Find the probability

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a w=9w kDae /
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Suggested solution for MATH 1853 homework 1
1. Determine the values of k so that the following system in unknowns x, y, z has:
(i) a unique solution, (ii) no solution, (iii) an infinite number of solutions:
x 2y 1
x y kz 2
ky 4 z 6
Solution:
Reduce the sy

13
Bernoulli Experiment and Its Related Distributions
An experiment is called a Bernoulli experiment (or Bernoulli trial) if there
are only two possible outcomes: success with probability p and failure with
probability (1 p), where 0 < p < 1.
We say X is

Example 1. Consider the function
cfw_ 2
3x if 0 x 1,
p(x) =
0 otherwise.
Determine if it is a probability density function. If yes find P (X 0.5).
1
We note that p(x) 0 for any x [0, 1]. Secondly, we see that
1
p(x)dx =
3x2dx = 1.
0
Therefore p(x) is a

MATH 1853
Problem Set #4
1.
1 2 0
Given A 2 2 0 , x
0 0 1
3
and x 1 .
(a) Decompose A into A QQT where Q is an orthonormal matrix and
is a diagonal matrix containing the eigenvalues of A . Then, decide if
A is positive definite, negative definite or

MATH 1853
Problem Set #3
1.
1 4 1
Find the eigenvalues and eigenvectors of the matrix A 2 1 1 .
3 1 2
2.
7 4 1
It is known that A 4 7 1 has eigenvalues
4 4
3,3,12 , find the value
of and the eigenvectors of A .
3.
Suppose the 3-by-3 matrix A has 3 ei

MATH 1853
Problem Set #2
1.
4
5 2
1 0 2
Given A 6 4 0 and B 3 1 2 .
1 4 3
6 2 1
Find A-1 and B-1. Verify that (AB)-1 = B-1A-1.
2.
1 2 3
Find all the cofactors of the determinant 1 0 1 . Hence evaluate the determinant.
1 1 1
3.
ab ac
Compute the determina

Definition of the Cross Product:
u v ( u v sin )n.
( now, it is a vector! )
The new vector n (the direction of the cross product) is perpendicular
to u and v. It is closely associated with the right-hand screw rule.
The construction of u v can be illustra

MATH 1853 Homework (Spring 2016)
1. Determine the values of k so that the following system in unknowns x, y, z has: (i) a
unique solution, (ii) no solution, (iii) an infinite number of solutions:
x 2y 1
x y kz 2
ky 4 z 6
2. Let W be the solution space of

MATH1853 II T6
Geo(p), Bin(n, p), Poi(), exp()
F. Tsang
University of Hong Kong
April 21, 2016
F. Tsang (University of Hong Kong)
MATH1853 II T6
April 21, 2016
1 / 12
Geometric random variables (Beginner)
Experiment: shooting at a target until there is a

MATH1853 II T5
Random variables
F. Tsang
University of Hong Kong
April 15, 2016
F. Tsang (University of Hong Kong)
MATH1853 II T5
April 15, 2016
1 / 12
Random variables
Recall: A sample space S is the collection of all possible outcomes of an
experiment.

Eigenvalues and Eigenvectors
For an nn matirx A, if there is a nonzero vector x such that Ax=lx
for some scalar l, then l is an eigenvalue of A, and x is the
eigenvector corresponding to l
If we view A as a mapping, Ax=lx means that the mapping A acting
o

Determinants
1 d b
ad bc c a
Recall the 22 matrix inverse equation A 1
Is it possible to extend the result to matrix of dimension nn ?
We first look at the concept of determinant
a11 a12
For 22 matrix A a a , its determinant is det(A)=a11a22-a12a21
2

Vectors
A real number is a point on the real line
To describe a point on a plane 2, we use two numbers, e.g., (3,-1)
In fact, this is just an expression of a point using rectangular
coordinate system
3
If we put the coordinates as , we have a vector
1
0

MATH1853 II T3
Complex variables:
roots of unity, complex functions
F. Tsang
University of Hong Kong
March 29, 2016
F. Tsang (University of Hong Kong)
MATH1853 II T3
March 29, 2016
1 / 11
Root of unity
How do we solve a polynomial equation
x 3 + ax 2 + bx

544 Answers to Selected Exercises
5. (a) y=2~ (b)y="4'f3 .
(c) For the ccrrelatinn to be equal to <1, the points wnzd have to lie on a straight line with negative 510113. There is
110 value for y far which this is the case.
7. (a) y = 337.1334 + 9.0980,:

Solutions for homework 1
1.1, #5
1 4 5 0 7
0 1 3 0 6
as the augmented matrix of
Consider the matrix
0 0
1 0 2
0 0
0 1 5
a linear system. State in words the next two elementary row operations that
should be performed in the process of solving the sys

MATH1853 Maths I
Eigenvalues & Eigenvectors
Dr. Ngai WONG
Fall 2015
1
Eigenvalues and Eigenvectors
For an nn matrix A, if there is a nonzero vector x such that Ax=x for
some scalar , then is an eigenvalue of A, and x is the eigenvector
corresponding to
I

The following proposition gives an alternative denition of the exponential distribution and is optional.
Proposition 1. A non-negative random variable X is exponentially distributed with parameter if and only if P cfw_X < t + h|X > t = h +
o(h) as h 0, wh

25 September 2015
Hi Students,
I have googled and compiled below a list of linear algebra sample questions and answers. I find it hard to
select sample questions on the internet one after one because there are numerous. Therefore, Ill advise
you to pick t

4
The Expectation of a Random Variable
Let X be a random variable with probability distribution function p(x).
The mean or the expected value E[X] of X is dened as
E[X] =
xp(x) if X is discrete,
x
and
E[X] =
xp(x)dx if X is continuous.
E[X] computes the l

(Dr. N. Wong 21.9.2015) An augmented matrix question from one of you, which I think I may just solve it as a
demonstration of this type of question involving pivot & free columns.
I use brackets to denote pivots for ease of typing. The steps should be obv

2.1
Some Complex Functions
In this section, we consider the image of a complex function.
2.1.1
Linear Function
Consider the linear function f : D C:
f (z) = cz + d where c, d R, c = 0.
The image of D under the function f is the set of complex numbers
f (D