Teaching Syllabus of Higher Algebra
【Course code】 JSZB0101
【The application of professional】 Mathematics and Applied Mathematics
【Class time】 180 Class hours
【Learning points】 8 Learning points
【Course nature, objectives and requirements】
This course is an important basic theory course of mathematical disciplines.Its
main task is to make students get the mathematics ideas and methods and the system
knowledge of polynomial theory, matrix theory, linear space theory, theory of linear
equations group and other aspects. It is a necessary foundation for studying modern
algebra, number theory, differential equations and other follow-up courses, is also a
mathematical tool which is widely used to natural science and engineering technology.
Also it plays an important role in improving students' abstract thinking and logical
reasoning ability, strengthening students' basic knowledge, basic theory and basic
skills and cultivating students' ability to solve practical problems.Through the
studying of this course, we make students to grasp some dialectical relationships such
as the concrete and abstract, special and general, finite and infinite and cultivate their
dialectical materialist point of view. Step by step culture students' abilities to
discovery and innovate the truth of knowledge and train their abilities to observate,
analysis, induction, synthesis, abstract summary and exploratly speculate. Also make
students deeply understand of middle school mathematics content, and can look down
from a height to grasp and handle the senior middle school mathematics teaching
material, and further improve the quality of mathematics teaching in secondary
school.
【Teaching schedule】
This course : 8 credits, 180 hours, time allocation as follows
Chapter
1
Course content
Collection, map, number ring and
number field etc.
Class
hour
Note ( Teaching )
Lectures and discussions,
14
exercises lessons, homework,
counseling
Polynomial definition, operation,
2
divisibility, factorization, the roots of a
Lectures and discussions,
28
polynomial
3
counseling
The definition, properties and
calculation of the determinant
Lectures and discussions,
16
method of solutions of the system of
Lectures and discussions,
12
linear equation
5
Matrix definition, arithmetic,
elementary matrix, invertible matrix
exercises lessons, homework,
counseling
The determination and calculation
4
exercises lessons, homework,
exercises lessons, homework,
counseling
Lectures and discussions,
18
exercises lessons, homework,
counseling
Lectures and discussions,
6
Vector space theory
26
exercises lessons, homework,
counseling
7
The definition and operation of Linear
transformations
Lectures and discussions,
26
exercises lessons, homework,
counseling
Lectures and discussions,
8
Euclidean space and unitary space
24
exercises lessons, homework,
counseling
Lectures and discussions,
9
The basic theory of quadratic forms
16
exercises lessons, homework,
counseling
Total
180
【Teaching contents】
Chapter 1
Basic concept
1. The purpose and requirement of learning
Basic requirements: To master the concepts of collection and mapping. Master
methods of Judging corresponds to mapping, surjection, single shot or double jet. To
prove propositions by applying the mathematical induction. Understanding the integer
divisible theory. Master number rings and number fields. To prove a number set to be
a number ring or a number field.
Higher requirements：Master proof methods of maps, surjections, single shots,
bijectives. Master ideas of using the maximum principle to prove the mathematical
induction theorem.
2. main teaching contents
1.1 Set
(1)The concept of set
(2)equal of set or subset
(3) Union, intersection, difference set;
(4) Operation laws of union and intersection
1.2 mapping
(1)The concept of mapping
(2) surjective, single shot, double shot
(3) Composition of mappings.
(4) Reversible mapping, necessary and sufficient condition of reversible
mapping;
1.3 Mathematical induction
(1) The natural numbers and the maximum principle
(2) The first mathematical induction
(3) The Second mathematical induction;
1.4 Divisibility of integers
(1) Definition of Divisibility
(2)Division with remainder
(3)The greatest common factor of integers
(4) Coprime of integers
1.5 number rings and number fields
(1)Concepts of number rings and number fields
(2) Any number field contains the rational field
Chapter 2
Polynomial
1. The purpose and requirement of learning
Basic requirements: Master the concept of polynomial over number ring R and
Division with remainder. Understand the concept and properties of the polynomial
divisibility. Skillfully proving divisibility of polynomial problem. Master the
definition, properties, calculating methods of the greatest common divisor of
polynomials and definitions and properties of coprime. Calculate the greatest common
divisor of two polynomials by using the Euclidean algorithm. Understanding the
concept of irreducible polynomials. Master the factorization theorem of polynomial.
Understanding concepts of the derivative of a polynomial and multiple factors. Master
the discriminant method of polynomials having no multiple factorization. Master
concepts
and
propertyes
of
polynomial
functions
and
polynomial
roots.
Understanding polynomial roots factoring problem over complex fields or real fields.
Master calculating methods of roots of polynomial over rational number fields. Use
Eisenstein discriminant method to determine the irreducible polynomial.
Higher requirements ： Master proof methods of division with remainder
theorem. Understand and grasp definitions and properties of the greatest common
divisor and the coprime of a system of polynomial. Master the proof method of the
unique factorization theorem. Master the equivalence of the equality of polynomial
and the equality of polynomial function.
2. main teaching contents
2.1 the definition and calculation of polynomial
2.2 divisibility of polynomial
2.3 the greatest common divisor of polynomials
2.4 the factorization theorem of polynomial.
2.5 multiple factorization
2.6 polynomial functions and roots
2.7 polynomial over complex fields or real fields
2.8 polynomial over rational number fields
Chapter 3
Determinants
1. The purpose and requirement of learning
Basic requirements: Understand definitions of the arrangement, exchange,
reverse order. Master calculation methods of inverse ordinal number. Understanding
the definition of determinants. Computate 2th order or 3th order determinant by using
diagonal ruler.Master calculation methods of common determinant ( triangular
determinant, Vandermonde determinant etc.).Grasp the properties of the determinant
and the determinant expansion theorem. Master some typical methods which are used
to calculate Vandermonde determinant such as the Vandermonde determinant method,
recursive method etc. Solve systems of linear equations by using Cramer algorithm.
Higher requirements：Understand deeply the concept of n order determinant.
Learn to calculate the various types of the value of determinant by using
Comprehensivly various types of methods.
2. main teaching contents
3.1 Systems of linear equations and Determinants
3.2 Arrangement
3.3 Definition and its basic properties of nth order determinants
3.4 minor and algebraic cofactor
expansion of determinants
3.5 Cramer rule
Chapter 4 Systems of linear equations
1. The purpose and requirement of learning
Basic requirements: Understanding the relationship between the elimination
method and the elementary transformation of matrix. Solve systems of linear
equations by proficiency using the elementary transformation of matrix. Understand
the concept and nature of the rank of matrix. Calculate the rank of matrix by
proficiency using the elementary transformation of matrix. Correctly understand
discriminant theorems of systems of linear equations having solutions, and can apply
it flexibly. Understand and grasp the theory of formula solutions of systems of linear
equations.
Higher requirements：Correctly understand the concept of the rank of matrix.
Grasp methods of proof of discriminant theorems of systems of linear equations
having solutions. Find solutions of systems of linear equations of parameters. Discuss
solutions of systems of linear equations of parameters. Grasp calculation methods of
formula solutions of systems of linear equations.
2. main teaching contents
4.1 Elimination method of systems of linear equations
4.2 Rank of matrix
4.3 Solving systems of linear equations
4.4 Conditions of Homogeneous systems of linear equations having non-zero
solution
4.5 Formula solution of systems of linear equations
Chapter 5
Matrices
1. The purpose and requirement of learning
Basic requirements: Understand the concepts of matrices. Know the differences
between matrices and determinants. Master definitions and properties of several
common matrices. Master the definitions and operation laws of matrix computation
and skillfully apply it. Understand the concept and nature of the reversible matrix.
Grasp the judgment methods of matrix inversion and calculating methods of inverse
of a square matrix. Grasp concepts and properties of the elementary transformation
and the elementary matrix. Understand the relationship between the two. Understand
the theorem of the determinant of matrix product and rank of matrix product and be
able to apply the theorem to prove relevant propositions. Comprehend and understand
the problem about the matrix block and elementary transformation of the matrix
block.
Higher requirements：To master the concept and nature of elementary matrices.
Master proof of the theorem of the determinant of matrix product and rank of matrix
product.
The determinant of matrix product and product of matrix rank theorem;
Solve some problems by using block matrix methods.
2. main teaching contents
5.1 The concepts and operations of matrix
5.2 The elementary transformation of matrix
5.3 Invertible matrix
5.4 The determinant and rank of matrix product
5.5 The matrix block computation
Chapter 6
Vector spaces
1. The purpose and requirement of learning
Basic requirements: Understanding the concept of vector space. Master the
simple nature of the vector space. Understanding the concept of subspace. Master
concepts of generated subspaces, summation subspaces and straight summation etc.
Be able to prove that a subset of a vector space is a its subspace. Understanding the
replacement theorem. Master definitions and properties of linear dependent and linear
independent. Understanding the definition of maximal linearly independent group. Be
able to calculate maximal linearly independent group of a vector group.
Understanding the concept of equivalence of vector group. Understand concepts and
natures of base, dimension and the coordinates. Be able to seek the basis,dimension
and the transition matrix of a vector space. Understanding definitions of isomorphism
mapping and isomorphism.
Master definitions of row rank of matrix and column
rank of matrix. Master concepts of solution spaces and the basic system of solutions
of homogeneous systems of linear equations. Be able to calculate the basic system of
solutions of homogeneous systems of linear equations.
Higher requirements：Deeply understand definitions of vector spaces and
isomorphism. Correctly understand concepts of summation subspaces and straight
summation. Master proof and application of the replacement theorem. Master the
equivalention of two definitions of matrix of rank. Be able to prove some inequalities
about rank.
2. main teaching contents
6.1 Definitions and simple properties of vector spaces
6.2 Subspace
6.3 linear correlation of vectors
6.4 Bases and dimensions
6.5 Coordinate
6.6 Isomorphism of vector spaces
6.7 Solution spaces of homogeneous system of linear equations
Chapter 7
Linear transformations
1. The purpose and requirement of learning
Basic requirements: Understanding the definition of the linear map.
Understanding the definitions and propties of the image and nuclear of the linear map.
Understanding the definition of the linear transformation. Master the operation and
basic properties of the linear transformation. Mastering the relationship between
linear transformation and matrix. Be able to prove similarity of matrices. Understand
the definition and properties of invariant subspaces. Deeply understand definitions
and properties of the eigenvalues and eigenvectors of of the linear transformation.
And grasp calculation methods of eigenvalues and eigenvectors. Understand and
grasp the theory and method that linear transformation ( matrix ) can be diagonalized.
Higher requirements：Be able to calculate the linear mapping image and
nuclear and their dimensions. Be able to prove the proposition about the image and
nuclear of the linear mapping. Understanding
the proof of isomorphic between L
（ V ） and M n ( F ) . Master proof of diagonalization theorem about linear
transformation and matrix.
2. main teaching contents
7.1 Concepts of linear mappings
7.2 Operators of linear transformations
7.3 Linear transformations and matrices
7.4 Invariant subspace
7.5 The eigenvalues and eigenvectors
7.6 Diagonalizable matrices
Chapter 8
Euclidean spaces and unitary spaces
1. The purpose and requirement of learning
Basic requirements: Understanding the definition of Euclidean spaces. Grasp
definitions and properties of the inner product of the vector, length, angle and
distance. Master the definition of standard orthogonal basis. Be able to calculate
standard orthogonal basis by using orthogonal method. Understanding the definition
and properties of orthogonal matrix. Master the definition, properties and decision
theorem of orthogonal transformation. Understanding the relationship among the
orthogonal transform and standard orthogonal basis and orthogonal matrix. Master the
definition,
properties
and
decision
theorem
of
symmetry
transformation.
Understanding the relationship among the symmetry transform and standard
orthogonal basis and symmetry matrix.Understanding the conclusion of symmetric
transformation and real symmetric matrix diagonalization. Understanding unitary
space and unitary transformation.
Higher requirements ： Master proof
methods of the Cauchy-Schwarz
inequality. Master the definition, properties and proof of the orthogonal complement.
Deeply understand orthogonal transformation types of two-dimensional and
three-dimensional space. Deeply understand the theory of unitary space. Master the
difference and connection between Unitary space and Euclidean space.
2. main teaching contents
8.1 Definitions of Euclidean spaces
8.2 Orthogonal bases
8.3 Orthogonal transformation
8.4 Symmetry transform
8.5 Unitary spaces
8.6 Unitary transformations and symmetry transforms
Chapter 9
Quadratic forms
1. The purpose and requirement of learning
Basic requirements: Understanding the definition of quadratic forms. Master
the method of denoting quadratic forms by using matrices. Understanding the
definition of rank of quadratic forms. Be able to calculate the rank and symbol
difference of quadratic forms. Understanding the definition and properties of the
contract matrix. Understanding the definitions of standard form and canonical form of
quadratic forms. Be able to simplificate quadratic forms into standard forms by using
the contract transformation method and the orthogonal transformation method.
Understanding the inertia theorem. Understanding the concept and nature of positive
definite quadratic forms. Be able to judge positive definite quadratic forms by using
master modes.
Higher requirements ： Understanding the difference and relation between
contract relationship and similarity relationship. Master inertial theorem. Deep ly
understanding proof of the spindle and its geometric significance. Master the
definition and proof of positive definite matrices. Understand the definitions and
determinations of the negative quadratic forms ( matrix ), semi-positive definite
quadratic forms ( matrix ), semi-negative definite quadratic forms ( matrix ),
indefinite quadratic forms ( matrix ) .
2. main teaching contents
9.1 Quadratic forms and symmetric matrices
9.2 Congruent transformation of matrices
9.3 Quadratic forms over complex number fields and real number fields
9.4 Positive definite quadratic forms
9.5 Spindle problems
【Performance assessment】
1. Evaluation of usual performance
Usual performance is evaluated according to the students' homework, class
attendance and quality assessment results.
2. Final evaluation
The final assessment practices closed book examination. Unifiedly set questions
according to the syllabus of unified proposition. Examination time for 120 minutes.
Roll surface score 100 points.
【Teaching materials and reference books】
Prescribed textbook:
Zhang He-rui, Hao Bing-xin.《Higher algebra》（Fifth Edition）, 2007,Higher
Education Press.
reference books:
1. Peng Guo-hua, Li De-lang. 《Linear algebra》, 2006,Higher Education Press.
2. Hao Zhi-feng,Xie Guo-rui,Wang Guo-qiang,Wu Zhi-jian. 《 Linear
algebra》,2009, Higher Education Press.
3. Bernard Kolman, David R.Hill. Wang Dian-jun adaptation. 《Linear Algebra
an Applied First Course》. 2005,Higher Education Press.
4. Steven J.Leon(American). Linear algebra (English Edition)(Sixth Edition).
2004, Machinery Industry Press.
5. David C.Lay.Linear algebra and Its Applications (English Edition)(Third
Edition). 2004,electronics industry Press.
【Related instructions（Teaching suggestions）】
1. Prerequisite courses ：The high school mathematics
2. 《Higher Algebra》is necessary foundation to《study modern algebra》,
《number theory》,《differential equations》and other follow-up courses.
3. Classroom teaching is main for this course. Exercises selected and exercise
class are supplemented. We pay attention to the basic concepts, basic theory of
infiltration and abstract thinking and train logical reasoning method.
（Organizate：Gao Xing-hui / Examine：Hou Wan-sheng）