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2022　Linear Algebra and Its Applications

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Academic unit or major
Undergraduate major in Mathematical and Computing Science
Instructor(s)
Sumita Hanna  Takabe Satoshi 
Class Format
Lecture / Exercise    (Livestream)
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(W323)  Fri3-4(W323)  
Group
-
Course number
MCS.T203
Credits
2
Academic year
2022
Offered quarter
1Q
Syllabus updated
2022/3/16
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

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.

Student learning outcomes

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 that will be developed

✔ 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/Required learning

  Course schedule Required learning
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 Assignment will be given
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

Out-of-Class Study Time (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.

Assessment criteria and methods

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 (i.e., required knowledge, skills, courses, etc.)

Preferred that credits for "Linear Algebra I / Recitation'', "Linear Algebra II'', and "Linear Algebra Recitation II'' are already obtained.

Other

(FY2022) The final exam will be given in-person during the examination period. Depending on the situation of the new corona virus, it may be given as an online examination or a final report.

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