2024 Faculty Courses School of Computing Undergraduate major in Mathematical and Computing Science
Linear Algebra and Its Applications
- Academic unit or major
- Undergraduate major in Mathematical and Computing Science
- Instructor(s)
- Hanna Sumita / Yu Yokoi
- Class Format
- Lecture/Exercise (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Tue / 3-4 Fri
- Class
- -
- Course Code
- MCS.T203
- Number of credits
- 110
- Course offered
- 2024
- Offered quarter
- 1Q
- Syllabus updated
- Mar 17, 2025
- Language
- Japanese
Syllabus
Course overview and goals
In the first part, a simple review of basic notions of linear algebra will be carried out focusing on the definitions. The comprehension of important notions such as linear independence/dependence of vectors and linear mapping will be confirmed through assignments involving their definitions. Also, some issues related to software implementations will be discussed through examples of rudimentary numerical methods to solve linear system of equations in order to understand more advanced methods of numerical analysis. In the second part, having in mind applications in engineering, development of basic notions of linear algebra to encourage better comprehension will be considered. For example, by reinterpreting the least square method using projections into vector subspaces. At the end, quadratic forms and eigenvalue problems of matrices, which are common notions always present in mathematics and computational mathematics, will be explained.
Course description and aims
Objective to attain: Master the notions of finite dimensional vector spaces, which is a basic concept in mathematics and numerical analysis, through assignments. Also comprehend the basic difficulties when applying these notions when solving linear system of equations and numerical methods involving matrices, in order to understand more advanced numerical methods.
Theme: Review the basic notions of linear algebra and to be able to not have difficulties in assignments involving them. Also, the connection to other lectures that need these notions will be considered and the attendees will be able to reinterpret these ideas in different contexts than linear algebra.
Keywords
N-dimensional Euclidean space, methods to solve linear system of equations, orthogonal projection, quadratic forms, eigenvalues and eigenvectors
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Definitions and theorems will be explained during the lectures according to the handouts, and assignments will be required almost at every lecture.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Introduction | Criteria to evaluate the comprehension, etc. |
| Class 2 | N-dimensional vector space: vector space, linear dependence/independence, subspaces, linear mapping, vector norms, matrix norms, inner products | Assignment will be given |
| Class 3 | Interpretations of linear system of equations | Assignment will be given |
| Class 4 | Determinants | Assignment will be given |
| Class 5 | Numerical methods for linear system of equations, inverse matrices, and determinants | Assignment will be given |
| Class 6 | Dimension, basis, and orthogonal complement of an n-dimensional vector space | Assignment will be given |
| Class 7 | Computation using Julia | |
| Class 8 | General assignment to check the comprehension | |
| Class 9 | Orthogonal projection (linear subspace, least square method) | Assignment will be given |
| Class 10 | Supplement issues for linear system of equations and operations (direct sum) involving linear subspaces | Assignment will be given |
| Class 11 | Quadratic forms and eigenvalues | Assignment will be given |
| Class 12 | Diagonalization of matrices and contour lines of quadratic forms | Assignment will be given |
| Class 13 | Complex matrices and Gershgorin theorem | Assignment will be given |
| Class 14 | Singular value decomposition | Assignment will be given |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None.
Reference books, course materials, etc.
Lecture notes will be distributed when necessary. A very basic reference is "Introduction to Linear Algebra, 4th edition", G. Strang, Wellesley Cambridge Press, 2009. A more advanced level reference is "Numerical Linear Algebra", L. N. Trefethen, D. Bau, III, SIAM, 1997.
Evaluation methods and criteria
Can answer questions which involve notions of n-dimensional vector spaces and understand the basic notions of numerical methods in linear algebra. Mid-term and final exam will count 80% of the grade and the remaining 20% comes from the assignments.
Related courses
- LAS.M102 ： Linear Algebra I / Recitation
- LAS.M106 ： Linear Algebra II
- LAS.M108 ： Linear Algebra Recitation II
Prerequisites
Preferred that credits for "Linear Algebra I / Recitation'', "Linear Algebra II'', and "Linear Algebra Recitation II'' are already obtained.