AY2015 NIT Ibaraki College
Shinzo BANNAI
Modern Mathematics II
Email: sbannai@ge.ibaraki-ct.ac.jp
Basic Information
Course
Advanced Course
Compulsory / Elective
Elective
Grade
2nd
Number of Credits
2
Semester
1st Semester
Credit Type
Special Credit Type 2
Class Type
Common Technical Subject
Course Objectives
This course is designed to relearn the concepts related to vectors, matrices and linear maps that were
learned during the Regular Course in a more abstract form. One of the aims is to become used to
considering and using abstract concepts through abstract vector spaces. Concepts related to bases,
matrices of linear maps and the Jordan normal form, which were not treated in the regular course, will be
treated. Also, the English of related mathematical terms will be introduced as needed. Furthermore, there
will be a reading assignment about real world applications of Linear Algebra. The goals of this course are:
1. To understand the concept of vector spaces and linear maps.
2. To understand the concept of a basis.
3. To understand linear maps and their matrices.
4. To understand “normal forms” of matrices in terms of linear maps and their matrices.
5. To become used to learning and using mathematics in English.
Topic Outline / Schedule
Week Topic
1
Vector Spaces
2
Linear dependence and
independence
Outline
Vector spaces, subspaces
Checking linear dependence and independence
3
The maximal number of linearly independent vectors
5
The maximal number of linearly
independent vectors
The basis and dimension of a vector
space
Linear maps
6
7
Linear maps and their matrices
Mid-term exam of 1st semester
Finding matrices of linear maps
8
Eigenvalues and eigenvectors
Choosing reading assignment
Eigenvalues, eigenvectors and Cayley-Hamilton’s
Theorem
9
Diagonalization
Progress report on assignment
Direct sums of vector spaces
Progress report on assignment
Minimal Polynomials
Presentation of assignment
Generalized eigenspaces
Diagonalization of matrices
4
10
11
12
The definition of a basis and dimension of a vector space
Linear maps
Direct sums
Minimal polynomials of matrices
Generalized eigenvectors and eigenspaces
1
AY2015 NIT Ibaraki College
13
The Jordan Normal Form
The Jordan Normal Form of nilpotent linear operators
14
15
The Jordan Normal Form
Final Exam on 1st semester
The Jordan Normal Form of general linear operators
16
Review of 1st semester
Textbooks and Other Readings
Text Book:
T. Miyake, “Linear Algebra, from basics to the Jordan Normal Form”, Baifuukan, (in Japanese)
Reference book:
H. Anton and C. Rorres, “Elementary Linear Algebra With Supplemental Applications”, Hoboken: John
Wiley & Sons
Relevant Online Resources
KHAN ACADEMY:
https://www.khanacademy.org/math
MIT Open Courseware: “Linear Algebra“, Professor Gilbert Strang
http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
Grading Policy
Grading will be based on the mid-term exam and the final exam and also assignments. The exams will
account for 70% of the grades and the assignments will account for 30%. You will need a score of 60 or
higher to earn credits.
Course Description
The material related to Calculus and Linear Algebra learned in the Regular Course are prerequisites. The
students are expected to prepare for and to review each class on their own. Also, they are expected to
solve as many problems as possible on their own. If there are any questions concerning the course, please
actively ask questions during class or utilize the Office Hours effectively.
2