The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(MATH111)[2010](s)midterm~cs_zxxab^_10440.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=9 at 20130831 00:28:14. Academic use within HKUST only.
1
HKUST
MATH111 Linear Algebra
Midterm Examination
25th March 20
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(MATH111)04SFinal.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=0 at 20130831 00:29:29. Academic use within HKUST only.
Math 111 Final Exam
May 21, 2004
Your Name
Student Number
Section Number
1. For more spa
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(math111)[2008](f)MATH111Q5~PPSpider^_10430.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=14 at 20130831 00:29:11. Academic use within HKUST only.
MATH111 Quiz 5A Solutions
Page 1/4
November 28, 2008
by Danie
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(math111)[2010](f)midterm~1406^_10436.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=17 at 20130831 00:29:22. Academic use within HKUST only.
MATH 111 Linear Algebra
L2: 18Oct2010,
Midterm Test, Fall 201011
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
Math 111E Midterm Exam, Fall 2004.
Q1. 1) Let A be a nonzero matrix with only one column, A’ be the transpose of A, what is the rank
of the square matrix AA”?
2) Find two noninvertible matrices whose product is invertibie.
3) Let T: V + W be a linear m
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
M A T H 31 21
(MATH2121)[2011](f)midterm~=7femuej^_20373.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH2121&id=1 at 20130830 23:22:55. Academic use within HKUST only.
(MATH2121)[2011](f)midterm~=7femuej^_20373.pdf
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
CHAP 1 SECTION 4: THE MATRIX EQUATION Ax = b
1. Section 4: The Matrix Equation Ax = b
Definition 1.1. Let A be an m n matrix with columns a1 , a2 , , an .
Suppose x is in Rn . The product of A and x is
x1
x2
Ax = a1 a2 an
= a1 x1 + a2 x2 + + xn an
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
CHAP 1 SECTION 5: SOLUTION SETS OF LINEAR
SYSTEMS
1. Homogeneous Linear Systems
Let A be a m n matrix and b is a vector in Rn . Then
Ax = b
is a linear system.
If b = 0 the system Ax = 0 is called homogeneous. It has at least one
solution x = 0 in Rn ,
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
CHAP 1 SECTION 2: ROW REDUCTION AND ECHELON
FORMS
1. Echelon Forms
Definition 1.1. A matrix is in echelon form (or row echelon form) if
All nonzero rows are above any rows of all zeros;
Each leading entry of a row is in a column to the right of the lead
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(math111)[2008](f)quiz3a~PPSpider^_10432.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=15 at 20130831 00:29:15. Academic use within HKUST only.
MATH111 Quiz 3A Solutions
Page 1/4
November 4, 2008
by Daniel Zh
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
Math 2121 Final Exam
May 19, 2014
Your Name:
Student ID Nnumber:
1. Show all your work. Cross off (instead of erase) the undesired part.
2. Provide all the details. Your reason counts most of the points.
Number
1
Score
2
3
4
5
6
7
Total
1
Math 2121 Final
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(MATH111)[2009](f)final~cs_ysx^_10433.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=4 at 20130831 00:28:49. Academic use within HKUST only.
Final Exam for Math 111 (L1), Dec. 19, 2009
Name: , ID: , Score:
Th
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(math111)[2008](f)MATH111Q2B~PPSpider^_10428.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=12 at 20130831 00:29:04. Academic use within HKUST only.
MATH111 Quiz 2A Solutions
Page 1/4
Oct 11, 2008
by Daniel Zh
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(math111)[2008](f)MATH111Q4~PPSpider^_10429.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=13 at 20130831 00:29:07. Academic use within HKUST only.
MATH111 Quiz 4A Solutions
Page 1/4
November 27, 2008
by Danie
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(math111)[2008](f)MATH111Q1~PPSpider^_10427.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=11 at 20130831 00:29:01. Academic use within HKUST only.
MATH111 Quiz 1 Solutions
1. (4 marks) Describe all the soluti
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(MATH111)[2010](f)quiz~1406^_10437.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=6 at 20130831 00:28:42. Academic use within HKUST only.
MATH 111 Linear Algebra
Quiz 2 for T2a
Name:
Student ID:
Time allowed:
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(MATH111)[2010](f)quiz~2247^_10438.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=7 at 20130831 00:28:38. Academic use within HKUST only.
MATH 111 Linear Algebra
Quiz 1 for T2a
Name:
Student ID:
Time allowed:
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(MATH111)[2008](f)quiz2a~PPSpider^_10431.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=3 at 20130831 00:28:53. Academic use within HKUST only.
MATH111 Quiz 2A
Page 1/3
Oct 11, 2008
by Daniel Zheng
. . . . . .
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
Math 2121 Final Exam
May 24, 2013
Your Name:
Student ID Nnumber:
1. Show all your work. Cross off (instead of erase) the undesired part.
2. Provide all the details. Your reason counts most of the points.
Number
1
Score
2
3
4
5
6
7
Total
1
1 (15 points) Co
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
(MATH111)[2010](s)final~cs_zxxab^_10439.pdf downloaded by mzhangag from http:/petergao.net/ustpastpaper/down.php?course=MATH111&id=8 at 20130831 00:28:32. Academic use within HKUST only.
FINAL EXAM MATH111 Linear Algebra Spring 2010
There are 5 problems
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
CHAP 1 SECTION 3: VECTOR EQUATIONS
1. Vector Equations
A matrix with only one column is called a column vector , or a vector.
Example 1.1. A real vector with two entries is
w1
w=
w2
where w1 , w2 are any real numbers.
Definition 1.2. The set of all vec
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
CHAP 1SECTION 1: SYSTEMS OF LINEAR EQUATIONS
1. Linear equation
A linear equation in the variables x1 , , xn is an equation that can be
written in the form
a1 x1 + a2 x2 + + an xn = b,
where b and the coefficients a1 , , an are real or complex numbers(co
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2016
CH 5 SEC 7: APPLICATIONS TO DIFFERENTIAL
EQUATIONS
1. Two examples
Let us use t to denote coordinate on real line R. We begin with the
following problem.
Let k be a real number. Can you find out all possible real valued
functions f = f (t) : R R that sati
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2016
CH 5 SEC 3: DIAGONALIZATION
1. Diagonalization
A square matrix A is said to be (real) diagonalizable if A is similar to a (real)
diagonal matrix, that is, if A = P DP 1 for some (real) invertible matrix P and
some (real) diagonal, matrix D.
Remark 1.1. I
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2016
CH 5 SEC 2: THE CHARACTERISTIC EQUATION
0.1. Determinanants. Let A be an n n matrix. Let U be any echelon form
obtained from A by row replacements and row interchanges (without scaling), and
let r be the number of such row interchanges.
Then the determin
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2016
CH 4 SECT 2: NUL, COL, AND LINEAR
TRANSFORMATIONS
1. Example of subspace of Rn : N ul and Col of a matrix
1.1. Subspaces of Rn .
Definition 1.1. A subspace of Rn is a subset H of V that has three properties:
The zero vector is in H.
H is closed under ve
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2016
CH 4 SECT 5:THE DIMENSION OF A VECTOR SPACE
1. The Dimension of a Vector Space
must
Theorem 1.1. If a vector space V has a basis B = cfw_b1 , , bn , then any set in
V containing more than n vectors must be linearly dependent.
Proof. Let cfw_u1 , , up be
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2016
CH 4 SECT 6: RANK
1. The Row Space
If A is an m n matrix, each row of A has n entries and thus can be identified
with a (horizontal) vector in Rn .
The set of all linear combinations of the row vectors is called the row space of
A and is denoted by Row
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2016
CH2 SECTION 2: THE INVERSE OF A MATRIX
1. Matrix operations
An n n matrix A is said to be invertible if there is a n n n matrix
C such that
CA = I
and AC = I
where I = In , the n n identity matrix.
In this case, C is an inverse of A.
In fact, C is uniq
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2016
CH 4 SECT 4: COORDINATE SYSTEMS
1. The Unique Representation Theorem
Theorem 1.1. Let B = cfw_b1 , , bn be a basis for a (real) vector space V .
Then for each x in V , there exists a unique set of scalars c1 , , cn such
that
span
(1.1)
x = c1 b1 + c2 b2
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2014
Information about Final Exam of Math 2121
The final exam will covers the following sections: Chapter 1. Sections 1,
2, 3, 4, 5, 7, 8. Chapter 2. Sections 1, 2, 3, 8, 9. (sections 8. 9 overlaps with
sections 1,2,3 of Chapter 4). Chapter 3. Sections 1, 2. C
The Hong Kong University of Science and Technology
Linear Algebra
MATH 2121

Fall 2016
CH2 SECTION 1 MATRIX OPERATIONS
1. Matrix Operations
Let A be an m n matrix.
The (i, j)entry of A, sometimes denoted by aij , is in the ith row and
jth column of A.
Each column of A is a list of m real numbers, ie a vector in Rm .
a11
:
ai1
:
a