Q1. Screws produced by a certain company
will be defective with a probability 0.01 independently of each other.
The company sells the screws in packages of
10 and oers a money-back guarantee that at
most 1 of the 10- screws is defective. What
proportion o
Q1. 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.
Answer. (a) |A| = 2 and |B| = 5.
(b)
A B = cfw_1 .
(c)
A B = cfw_1, 2, 3, 5, 6, 7 .
(d) The subsets of A are
, A ,
Q1. Andrew, Beatrix and Charles are playing
with a crown. If Andrew has the crown, he
throws it to Charles. If Beatrix has the crown,
she throws it to Andrew or to Charles, with
equal probabilities. If Charles has the crown,
he throws it to Andrew or to B
Q1. Let z = 1 + i. Find |z| and Arg(z).
Answer.
|z| =
12 + 12 =
2 .
and
Arg(z) = tan1
1
= .
1
4
1
Q2. Verify that each of the two numbers
z = 1 3i
satises the equation
z 2 2z + 4 = 0 .
Answer. If z = 1
3i, then we have
z 2 2z + 4
2
= (1 3i) 2(1 2 3i) +
3
Complex Variables
The imaginary number 1 i is a solution of the equation:
x2 + 1 = 0.
This idea of i was introduced to answer the above question. But then it results
in many interesting results, beautiful theory and useful applications.
In general a c
Inner Product
For two vectors u, v in n, the inner product is defined as
v1
v
uT v u1 u2 . un 2 u1v1 u2v2 . un vn
vn
It is obvious that uTv= vTu
From inner product, we can define other attributes of a vector
The length (or norm) of a vector v
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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
Matrices
As seen in the previous chapter, a matrix consists of vectors as
columns: A=[a1 a2 an]
For an mn matrix, its structure is
Two matrices are equal iff they have the same size and their
corresponding entries are equal
Sum of two matrices is just the
MATH1853 II T3
Complex variables: roots of unity
Counting & probability
F. Tsang
University of Hong Kong
April 7, 2017
F. Tsang (University of Hong Kong)
MATH1853 II T3
April 7, 2017
1 / 18
Root of unity
How do we solve a polynomial equation
f (x) = x 2 +
MATH1853 II T2
Complex variables
F. Tsang
University of Hong Kong
March 27, 2017
F. Tsang (University of Hong Kong)
MATH1853 II T2
March 27, 2017
1 / 15
Complex Numbers
z = a + bi
(a, b R, i =
1) ,
Re(z) = a ,
Im(z) = b
(not bi) ,
2
2
|z| = a + b .
Simila
True/False Questions
System of linear equations
If a system of linear equations has two different
solutions, it must have infinitely many solutions
If
a11 x1 a12 x2
a1n xn
b1
a21 x1 a22 x2
a2 n xn
b2
am1 x1 am 2 x2
amn xn
True
has at least two different
Linear Algebra
a x
a x
a n xn
b
a x
a x
a n xn
b
am x
am x
amn xn
bm
a
a
a
a
an
an
x
x
b
b
am
am
amn
xn
bm
m
A
m nn
m
A
A
a
a
a
a
a
a
a
a
a
an
an
an
am
am
am
amn
x
x
x
x
x
x
x
x
x
x
A
Solution
a
a
a
a
n
b
a
a
a
a
n
b
a
a
a
a
n
b
x
form
row- echelon form
x
Inner Product
For two vectors u, v in
n
, the inner product is defined as
v1
T
u v
u1 u2 . un
v2
u1v1 u2v2 . un vn
vn
It is obvious that uTv= vTu
From inner product, we can define other attributes of a vector
The length (or norm) of a vector v is defi
A Review on Set Theory and Basic Calculus
1
Set Theory
This section aims to introduce some fundamental notions in set theory, which are
often used in all branches of mathematics.
Definition 1. A set is a well-defined collection of objects.
The keyword her
4
The Variance of a Random Variable
Let X be a random variable with probability distribution p(x). The variance of
X, denoted by Var(X), is defined as
!
" #
2
V ar(X) = E (X E[X]) =
(x E[X])2 p(x) if X is discrete
x
and
!
"
V ar(X) = E (X E[X])2 =
$
(x E[
Quadratic Forms
We can generate a quadratic form starting with any n n matrix. For
example, let
2 3
A=
,
5 6
In the study of linear algebra, we have been concerned with linear
expressions, such as
3x 5y ,
2x + 4y 9z,
and for x = (x1 , x2 ), form the produ
Cross Product
Recall we have seen the inner product (or called the dot product),
with definition
v1
v
u v u v u1 u2 . un 2 u1v1 u2v2 . un vn
vn
T
Notice that the input of this operation is two vectors (of same
dimension), and the output is a sca
True/False Questions
System of linear equations
If a system of linear equations has two different
solutions, it must have infinitely many solutions
If
a11 x1 a12 x2 a1n xn b1
a x a x a x b
21 1 22 2
2n n
2
am1 x1 am 2 x2 amn xn bm
has at least two diff
AS/1853/MATH1853/3
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Assignment 3 Probability Distributions
Due Date: April 21, 2017 (FRI) (23:59)
1. (a) Find the probability of having more girls than boys in a family of 8 children.
(b) Find also
AS/1853/MATH1853/2
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Assignment 2
Due Date: April 14, 2017 (FRI) (23:59)
1. Find the 5 complex roots of z 5 = 2 2i.
2. In order to enter into a competition, 8 people have to be divided into three gr