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.
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.
N-dimensional Euclidean space, methods to solve linear system of equations, orthogonal projection, quadratic forms, eigenvalues and eigenvectors
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
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 | |
---|---|---|
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.
Lecture notes will be distributed when necessary.
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.
Preferred that credits for "Linear Algebra I / Recitation'', "Linear Algebra II'', and "Linear Algebra Recitation II'' are already obtained.
Mituhiro Fukuda (mituhiro[at]is.titech.ac.jp)
None.