### 2019　Linear Algebra and Its Applications

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Undergraduate major in Mathematical and Computing Science
Instructor(s)
Fukuda Mituhiro  Yamashita Makoto
Course component(s)
Lecture / Exercise
Day/Period(Room No.)
Tue3-4(W833)  Fri3-4(W833)
Group
-
Course number
MCS.T203
Credits
2
2019
Offered quarter
1Q
Syllabus updated
2019/3/18
Lecture notes updated
2019/6/8
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 foment 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 considered.

### 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

 Intercultural skills Communication skills ✔ Specialist 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 part I: vector space, linear dependence/independence, subspaces, linear mapping Assignment will be given
Class 3 N-dimensional vector space part II: vector norms, matrix norms, inner products Assignment will be given
Class 4 Interpretations of linear system of equations Assignment will be given
Class 5 Numerical methods for determinants and linear system of equations Assignment will be given
Class 6 Numerical methods for linear system of equations and inverse of matrices Assignment will be given
Class 7 Dimension, basis and orthogonal complement of an n-dimensional vector space Assignment will be given
Class 8 General assignment to check the comprehension
Class 9 Computation using MATLAB Assignment will be given
Class 10 Orthogonal projection (linear subspace, least square method) Assignment will be given
Class 11 Supplement issues for linear system of equations, operations (direct sum) involving linear subspaces Assignment will be given
Class 12 Quadratic forms and eigenvalues Assignment will be given
Class 13 Contour lines of quadratic forms and diagonalization of matrices Assignment will be given
Class 14 Numerical methods for eigenvalues Assignment will be given
Class 15 Complex matrices and applications of matrices Assignment will be given

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. 