Qatar University
College of Arts and Sciences
Department of Mathematics and Physics
Linear Algebra , SYLLABUS
COURSE INFORMATION:
 Course Number: 231
 Course title: Linear Algebra (1)
 Course Hours: 3 (2+2)
 Prerequisites: Calculus (1)
COURSE OBJECTIVES:




. To introduce systems of linear equations and discuss methods to solve them.
To introduce matrices and their properties.
To acquaint students with determinants and their properties and applications.
To familiarize students with vector spaces.
To learn about linear transformations.To learn about
Eigenvalues and Eigenvectors.
INSTRUCTOR:
Dr. huda Al Thani
 E-mail: h.m.althani@qu.edu.qa
 Office : 485-1624
 Location: SB208
OFFICE HOURS:
 12 -2 Sunday
 12 -2 Thursday
 Or by Appointment
I strongly encourage you to take advantage of my office hours.
EVALUATION POLICY:
1
Three major exams will be given:
 First Exam: 25%,
 Second Exam: 25%,
 Final Exam: 40%
 Quizzes: 10%
INSTRUCTIONS & REGULATIONS:
 Using Mobile phones during lectures or exams is not allowed.
 Students are expected to attend at least 75% of the classes, otherwise they
fail the course. No grades for attendance.
 No make ups on quizzes.
 Students are expected to participate actively in the class.
 Made up tests cannot be arranged except in case of emergency or absence
due to official university business.
 Check Your e-mail regularly
 Check dohamath.com regularly
 Come and see me as soon as you have questions
 If you are a student with special need, Please inform the professor. Then,
arrangements can be done with the Special Needs Section at the university
SYLLABUS ITEMS:






Systems of Linear Equations:
Introduction to systems of linear equations. Row operations. Gaussian eliminations.
Gauss-Jordan eliminations. Homogeneous linear systems.
Matrices and Matrix Operations:
Elementary matrices. The inverse of a matrix. Existence of the inverse. The inverse as a
product of elementary matrices. The inverse via row operations.
Determinants:
Definition and properties. The existence of the inverse of a matrix. The minor, cofactor
and adjoint. Cramer’s rule.
Vector Spaces:
Vector spaces. Sub vector spaces. Linear independence and linear dependence. The
spanning set of a vector space. Basis. Dimension. Row space. Column space. Null space.
Rank of a matrix. Nullity of a matrix.
Linear Transformations:
Linear transformation. Kernel. Range. Inverse of a linear transformation. Existence of the
inverse transformation. Determining the rule of the inverse. Linear transformations and
matrices.
Eigenvalues and Eigenvectors:
The transition matrix from a basis to another. Eigenvalues, and Eigenvectors.
Diagonalization. Applications.
2

Course Outline:
(The number after each section is the approximate number of class period)
Chapter 1: Systems of linear equations and matrices
1.1
1.2
1.3
1.4
1.5
1.6
Introduction to system of Linear equations (1)
Gaussian Elimination (2)
Matrices and Matrix operations (2)
Inverse: Rules of Matrix Arithmetic (1)
Elementary matrices and a method for finding A1 (2)
Further results on systems of equations and invertiblity (1.5)
Chapter 2: Determinantes
2.1
2.2
2.3
2.4
The determinante function (1)
Evaluating determinants by brow reduction (1)
Properties of the determinate function (1)
Cofactor expansion; Cramer's Rule (1)
Chapter 5: General vector spaces
5.1
5.2
5.3
5.4
5.5
5.6
Real vector spaces (2.5)
Subspaces (4)
Linear independence (2.5)
Basic and dimension (4)
Row space, column space and Null space (4)
Rank and Nullity (2)
Chapter 6: Inner product spaces
6.5
Change of Basis (1.5)
Chapter 7: Eigenvalues, Eigenvectors
7.1
7.2
Eigenvalues and eigen vectors (2.5)
Diagonalization (2.5)
Chapter 8:Linear Transformations
8.1
8.2
8.3
8.4
8.5
General linear Transformations (3)
Kernel and Range (3)
Inverse linear transformations (3)
Matrices of General linear transformations (3)
Similarity (2)
3
Learning outcomes
The students are expected to be able to:
Objectives
Learning Outcomes
THE STUDENT WOULD BE ABLE TO:
A. To introduce A.1.
systems of linear
equations
and
discuss methods to A.2.
solve them.
Use elementary row operations to solve systems of linear
equations.
A.3.
Discuss possible solutions of systems of linear equations and
introduce the vocabulary related to it.
A.4.
Use the Gaussian eliminations and backward substitutions to
solve systems of linear equations.
A.5.
B. To introduce B.1.
matrices and their
B.2.
properties.
C.
To
acquaint
students
with
determinants
and
their properties and
applications.
Define systems of linear equations and use augmented matrices
to describe them.
Use the Gaussian-Jordan eliminations to solve systems of linear
equations.
Identify matrices and know matrix properties.
Identify invertible matrices and know the rules of matrix
arithmetic.
B.3.
Know elementary matrices and use them to find the inverse.
B.4.
Discuss further rules on systems of linear equations and
invertibility.
B.5.
Identify diagonal, triangular and symmetric matrices and know
appreciate their properties.
C.1.
Produce the definition of the determinant function.
C.2.
Evaluate determinants by row reduction.
C.3.
Recognize the properties of the determinant function.
C.4.
Compute determinants using the cofactor expansion.
Determine the inverse of an invertible matrix using the
determinant and adjoint.
Apply Cramer’s rule in find solutions of systems of linear
equations.
Comprehend vector and subvector spaces.
C.5.
C.6.
D. To familiarize D.1.
students with vector D.2.
spaces.
D.3.
D.4.
Recognize linear independence and linear dependence.
Know the spanning set, basis, and dimension of a vector space.
Comprehend the row and column spaces, rank and nullity of a
matrix.
4
E. To learn about E.1.
linear
E.2.
transformations.
E.3.
Know general linear transformations.
Find out the existence of the inverse transformation, and know
some methods to find it.
Know the matrices of general linear transformations.
F. To learn about F.1.
Eigenvalues
and F.2.
Eigenvectors.
F.3.
Compute the transition matrix from a basis to another.
Find the Eigenvalues and Eigenvectors of a matrix.
Comprehend the matrix diagonalization problem and find
procedure to diagonalize a matrix.
Use diagonalization in geometric and algebraic multiplicity.
F.4.

Homework
Recommended Problems in the Textbook, to be attempted by the students
Chapter
#
Section
#
1.1
1.2
Page
#
6
19
Problems #
33
47
56
46
1(a,b,c),2,3,(a,d),4 (a,c)5,(a,b),6(a),7,8,11,13
1,2,3,4, 5, 6, 7, 8 (a, b),9,10 (a, b) ,11, 12, 13 (b, c), 14 (a, c),
15,17, 18, 19, 20, 22, 24, 25
2,3 (a,d, c), 5 (a, b), 13, 14
4,7 (a, d)
1, 2, 3, 4, 5 (a, c), 6 (a, b), 7 (a, c, d) 8, 9, 10, 11, 12, 13, 19
1, 4, 6, 7, 9, 12, 15, 16, 17, 20, 21, 22
74
1, 2, 6, 7, 8, 10
2.2
2.3
2.4
5.1
5.2
89
102
113
209
219
5.3
5.4
5.5
5.6
229
243
257
271
1, 2, 3, 4, 6, 9, 11, 12, 13, 14 (a),
1 (a), 2, 3, 5, 6, 7, 8, 10, 12, (b), 13, 17 (a), 20, 22
1, 3, 4, 5, 7, 10, 11, 13, 15, 16, 19, 21, 22, 23
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,21,22,24,28,29
1,2,3,4,5 (c, d), 6 (a, d, f),7,8,9,10 (a, b), 11, (a, b), 12, 13, 14, (b,
c), 19, 23
1,2,3(a, c),4 (a, d), 5(b), 7,8,10,12,14,15,17,18,20(a, c), 25
1,2,3,4,5,6,7,8,9,10,11,13,14,15,17, (a, c) ,18, 20, 21, 22, 26,35
1,2 (a, b, d, c),4,5(a, b, c), 6 (a, b, d) , 7, 8, 9, 10, 11, 12, 13,14, 15
1,2 (a, b, d), 3, 4, 5, 6, 8, 9, 10, 11, 12 (a), 13,
3,4,5,8,9
230
344
354
373
380
388
5 (a), 6 (a),7, 8, 9, 10, 12,, 14, 15
1,2, 3, 4, (b,d, f), 5(b, d, f),6 (b,d,f), 7(a)8(a), 9(a), 10, 11, 14,24
2,3,6,7,9,10,13, 14, 15, 16, 20, 21
1, 2, 5,7, 8, 9,12, 13, 15, 16, 17 (a,d),22, 27
1, 2, 5, 6, 7, 8, 9, 10,12,15, 16, 18,24, 25
1,2, 3, (a, d) 4, (a, c ), 5, 6, 7, 8, 10, 15, 17, 22, 24
1
1.3
1.4
1.5
1.6
Suuptement
ary
2
5
suuplement
ary
6
7
8
6.5
7.1
7.2
8.1
8.2
8.3
5
8.4
8.5
Suuplement
ary
399
411
413
1, 2, 3,5, 6, 8, 9,11,13, 16, 18
1, 3, 4, 5, 7,
1, 13, 14, 17, 20
6