MATH1853 Maths I
System of Linear Equations
Dr. Ngai WONG
Fall 2014
System of linear equations
A simple example: x 2 x = 1
1
2
x1 + 3 x2 = 3
Physical meaning: Solution is
the intersection point of two
lines
Other possibilities: No solution
or infinitely
MATH1853 Maths I
Matrix/Matrices
Dr. Ngai WONG
Fall 2014
1
Matrices
As seen in previous notes, 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
correspond
MATH1853 Maths I
Vectors
Dr. Ngai WONG
Fall 2014
1
Vectors
A real number is a point on the real line R
To describe a point on a plane R2, we use two numbers, e.g., (3,-1)
In fact, this is just an expression of a point using rectangular or
Cartesian coordi
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Point Estimate
A point estimate is a single number, calculated from the obtained sample data,
which is used to estimate the value of an unknown population parameter.
17.1
Point Estimate of the Population Mean
Very often, the mean of a population is an
13
Bernoulli Experiment and Its Related Distributions
An experiment is called a Bernoulli experiment if there are only two possible
outcome: success with probability p and failure with probability (1 p) where
0 < p < 1.
We say X is a Bernoulli random vari
TUT/1853/MATH1853/3
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Tutorial 3
1. Let 3 = 1 and = 1. Find the value of 2011 + 1997 + 1.
2. Show that tanh(ix) = i tan(x).
3. From a group of 5 women and 7 men, how many dierent committees of 2 wom
TUT/1853/MATH1853/4
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Tutorial 4
1. 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
Ch
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
MATH1853 Maths I
Inner Product
Dr. Ngai WONG
Fall 2014
1
Inner Product
For two vectors u, v in R , the inner product is defined as
n
v1
v
uT v = [u1 u2 . un ] 2 = u1v1 + u2 v2 + . + un vn
vn
It is obvious that uTv= vTu
From inner product, we can d
MATH1853 Maths I
Eigenvalues & Eigenvectors
Dr. Ngai WONG
Fall 2014
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
MATH1853 Maths I
Determinant
Dr. Ngai WONG
Fall 2014
1
Determinants
1 d b
Recall the 22 matrix inverse equation A =
ad bc c a
Is it possible to extend the result to matrix of dimension nn ?
We first look at the concept of determinant
a a
A = 11 12 , its
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Fall 2013
MATH1853 (Linear Algebra)
Dr. N. Wong
Problem Set #1
1. Solve the following equations by Gaussian elimination.
2x1 + 3x2 + 11x3 + 5x4 = 2
x1 + x2 + 5x3 + 2x4 = 1
.
2x1 + x2 + 3x3 + 2x4 = 3
x1 + x2 + 3x3 + 4x4 = 3
2. When a and b are real numbe
TUT/1853/MATH1853/2
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Tutorial 2
1. Let z = 1 + i. Find |z| and Arg(z).
2. Verify that each of the two numbers z = 1
3i satises the equation
z 2 2z + 4 = 0.
3. Reduce the following quantity to a re
TUT/1853/MATH1853/5
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Tutorial 6
1. Suppose that X follows N (3, 9), nd
(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 t
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
Gamma Distribution
1. Which of the following expression is equivalent to (5)?
a. 5
b. 5
c. 0 4
d. 4
e. None of the above
2. Find the value of (5).
a. 5
b. 24
c. 25
d. 120
e. None of the above
3. Find the value of (2.5).
a. 0.75
b. 1.33
c. 2.36
d. 2.5
MATH1853 Linear Algebra, Probability and Statistics Homework 1
Due Date: 12 Oct 2016 (Wed) Office Hours. Hardcopy Only. Please put into HW
collection box at Chow Yei Ching 601 (EEE Reception Counter) and sign on name list.
1.
Solve the following two syste