到達目標
1. To understand vector spaces and linear transformations.
2. To understand the concept of a basis.
3. To understand and to be able to calculate representation matrices.
4. To understand Jordan's normal form.
5. To become able to study mathematics in English.
ルーブリック
| 理想的な到達レベルの目安 | 標準的な到達レベルの目安 | 未到達レベルの目安 |
| 評価項目1 | The student understands the concepts of abstract linear algebra well and can conduct calculations in the abstract setting. Furthermore, the student can explain the concepts in ones own words. | The student understands the concepts of abstract linear algebra and can conduct calculations in the abstract setting. | The student does not understand the concepts of abstract linear algebra. |
| 評価項目2 | The student knows an application of linear algebra well and gives a clear and concise presentation. | The student knows an application of linear algebra and gives a satisfactory presentation. | The student does not know any applications of linear algebra. |
学科の到達目標項目との関係
教育方法等
概要:
This class will be held in English.
In this class, we will revisit various concepts treated in the regular course (such as vectors, matrices, linear transformations) and relearn them in an more abstract setting. The goal of this course is to learn how to think and get used to thinking in an "abstract setting" through studying "abstract linear algebra". We will also learn more deeply about bases of vector spaces, representation matrices, diagonalization and Jordan's normal form.
Furthermore, we will study various applications of linear algebra to real life, and each student will be required to make a presentation about an application.
The lectures will be given in normal classroom style, but if the number of registered students is small, it may be changed to a seminar style class.
In that event, the presentation will be canceled and the grading scheme will also be changed accordingly.
授業の進め方・方法:
The prequisites for this course are all of the materials treated in the classes of " 基礎数学I, II", "解析学", "代数・幾何" of the regular course. Each student is required to study on his own at home and should participate actively in each lecture. The students are advised to make use of office hours effectively and ask many questions both in class and at office hours.
注意点:
授業計画
|
|
週 |
授業内容 |
週ごとの到達目標 |
| 前期 |
| 1stQ |
| 1週 |
Vector spaces |
Vector spaces and subspaces. The condition for a set to be a subspace.
|
| 2週 |
Linear dependence and independence |
Linear dependence and independence. To be able to determine Linear dependence and independence.
|
| 3週 |
The maximum number of linearly independent vectors |
To be able to find the maximum number of linearly independent vectors for a give set of vectors.
|
| 4週 |
Basis and dimension of a vector space |
Understand the concept of dimension for an abstract vector space. To be able to find a basis and/or determine the dimension of a vector space.
|
| 5週 |
Linear transformations |
The definition of a linear transformation. Able to determine if a transformation is linear or not.
|
| 6週 |
Representation matrix of a linear transformation |
The definition of a linear transformation. To be able to find the representation matrix of a linear transformation for a given basis.
|
| 7週 |
（中間試験） |
|
| 8週 |
Eigen values and Eigen vectors
Choosing the topic of presentation |
The definition of eigen values and eigen vectors. To be able to calculate them.
|
| 2ndQ |
| 9週 |
Diagonalization
Interim report of presentation |
To be able to understand the process of diagonalization.
|
| 10週 |
Direct sum of vector spaces
Interim report of presentation |
The definition of a direct sum. To be able to determine if a sum of vector spaces is direct or not.
|
| 11週 |
Characteristic polynomials and minimal polynomials
Interim report of presentation |
The definition of minimal polynomials. To be able to calculate characteristic polynomials and minimal polynomials.
|
| 12週 |
Generalized eigen spaces
Interim report of presentation |
The definition of generalized eigen spaces. To be able to find a basis for generalized eigen spaces.
|
| 13週 |
Jordan's normal form (1) |
Jordan's normal form for nilpotent matrices.
|
| 14週 |
Jordan's normal form (2) |
Jordan's normal form for general matrices.
|
| 15週 |
（期末試験） |
|
| 16週 |
Presentation |
Presentation about applications of linear algebra in front of class.
|
評価割合
| 試験 | 発表 | | | | | 合計 |
| 総合評価割合 | 70 | 30 | 0 | 0 | 0 | 0 | 100 |
| 講義形式の場合 | 70 | 30 | 0 | 0 | 0 | 0 | 100 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |