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課程名稱 |
線性代數導論二
Introduction to Linear Algebra (II) |
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開課學期 |
112-2 |
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授課對象 |
數學系 |
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授課教師 |
蔡國榮 |
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課號 |
MATH4022 |
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課程識別碼 |
201E49990 |
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班次 |
01 |
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學分 |
4.0 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期三3,4(10:20~12:10)星期五3,4(10:20~12:10) |
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上課地點 |
普102普102 |
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備註 |
本課程以英語授課。數學系學生修課不計入畢業學分。輔數學系的學生得替代線性代數二。
總人數上限：100人 |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
This is a second course in linear algebra. In this course, three main topics will be discussed in details.
The first main topic is on Jordan canonical form and its related results. For an endomorphism T on a complex vector space of finite dimension, neither its geometric multiplicities nor its minimal polynomial is sufficient in characterising it up to conjugation by invertible matrices. The theory of Jordan canonical form resolves this problem completely and sheds new light on our understanding of linear operators that are not necessarily diagonalisable.
Secondly we treat (real or complex) vector spaces endowed with an appropriately defined “inner product”. These structures are ubiquitous and fundamental in mathematics and many parts of the sciences. We will cover orthonormal basis, Gram-Schmidt process, spectral theorems, singular value decomposition (SVD), Moore–Penrose inverses and their applications , among other things.
In the final part of the course, we will discuss some contemporary topics in linear algebra. We will focus on matrices with non-negative entries. As applications, we will discuss Markov chain, stochastic matrices and prove the Perron-Frobenius theorem which leads to the mathematics behind Google.
The syllabus can be found below.
(Part III - Jordan Canonical Form)
Week 1. Jordan Canonical Form (I) : The statement, examples and applications
Week 2. Jordan Canonical Form (II) : Jordan chains and generalized eigenspaces
Week 3. Jordan Canonical Form (III) : The proof
(Part IV - Inner Product Spaces)
Week 4. Inner product space (I) : definition and examples
Week 5. Inner product space (II) : Gram-Schmidt process
Week 6. Adjoint operators : normal, unitary, self-adjoint
Week 7. Spectral Theorem : statement and proof
(Part V - Applications of Linear Algebra)
Week 9. Singular value decomposition (SVD)
Week 10. Moore-Penrose's inverse
Week 11. Bilinear and quadratic forms
Week 12. Definiteness of matrices, Second derivative tests
Week 13. Markov chain
Week 14. Perron-Frobenius Theorem and mathematics behind Google
Week 15. Representation theory of finite groups
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課程目標 |
After finishing this course, students are expected to
1. be able to compute the Jordan canonical form of a given endomorphism over a complex vector space;
2. be familiar with finite-dimensional inner product spaces to a very general extent, both computationally and theoretically
3. be able to state and prove Spectral Theorems for normal and symmetric operators of an inner product space;
4. have a working knowledge of Markov chain and stochastic matrices and understand the mechanism behind the page rank algorithm of Google. |
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課程要求 |
Students are expected to have taken a first course in linear algebra, such as MATH4018 (Intro. to Linear Algebra 1) or linear algebra courses offered by other departments.
To be specific, we expect students to have known a reasonable set of vocabulary and fundamental results in linear algebra. For example,
- Vector space over a field
- Linear maps between vector spaces and their correspondence with matrices
- Kernel and image of a linear map, Rank and nullity theorem
- Change of coordinate matrices
- Eigenvalues, eigenvectors and eigenspaces
- Determine whether a given square matrix (over C) is diagonalizable |
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預期每週課後學習時數 |
Besides the 4-hour lectures per week, students should expect to spend around 2-3 hours weekly in digesting the lecture materials as well as completing exercises offered by the lecturer or the teaching assistant(s). |
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Office Hours |
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指定閱讀 |
We will be using materials from
1. S. H. Friedberg, A. J. Insel, L. E. Spence, "Linear Algebra", 4th Edition, Pearson Education, 2014 ; ISBN, 0321998898.
2. H. Dym, "Linear Algebra in Action" |
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參考書目 |
These books would also be useful, for example,
1. K. Hoffman and R. Kunze, "Linear Algebra".
2. Herstein and Winter, "Matrix theory and linear algebra". |
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評量方式
(僅供參考) |
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針對學生困難提供學生調整方式 |
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上課形式 |
以錄影輔助 |
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作業繳交方式 |
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考試形式 |
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其他 |
由師生雙方議定 |
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週次 |
日期 |
單元主題 |
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第1週 |
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1.1 Some reviews of MATH4018
1.2 Jordan blocks
1.3 Direct sum of matrices
1.4 Jordan Canonical Form
1.5 JCF Theorem |
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第2週 |
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2.1 Jordan chains : Definition
2.2 A worked example
2.3 Generalized eigenspaces
2.4 Jordan decomposition theorem |
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第3週 |
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3.1 Further properties of generalized eigenspaces
3.2 Proof of Jordan decomposition theorem
3.3 Application : Power of a matrix (revisit) |
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第4週 |
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4.1 Reviews on ‘dot product’ on C^n
4.2 (General) Inner product spaces
4.3 Properties of inner product spaces
4.4 Orthogonality and orthonormality
4.5 Orthonormal basis |
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第5週 |
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5.1 Gram-Schmidt process
5.2 Orthogonal projection
5.3 Orthogonal complement
5.4 Linear regressions |
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第6週 |
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6.1 Adjoint operators
6.2 Special operators and matrices
6.3 Real story : orthogonal and symmetric matrices
6.4 Spectral Theorems |
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第7週 |
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7.1 Proof of Complex Spectral Theorem
7.2 Use of Spectral Theorems |
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第8週 |
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Midterm Exam Week |
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第9週 |
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9.1 Statement and proof of SVD
9.2 A worked example of SVD
9.3 Polar decomposition
9.4 Applications : Low-rank approximation
9.5 Von Neumann’s trace inequality
9.6 Proof of Low-rank approximation |
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第10週 |
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10.1 Pseudoinverse : Theoretical definition
10.2 Pseudoinverse : Computational aspect via SVD
10.3 Penrose conditions
10.4 Applications : Regression method revisited |
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第11週 |
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11.1 Bilinear form : Definition
11.2 Matrix representations of a bilinear form
11.3 Symmetric bilinear forms
11.4 Quadratic forms
11.5 Sylvester’s law of inertia
11.6 Proof of Sylvester’s law of inertia
11.7 Review : Second derivative tests for f(x, y) |
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第12週 |
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12.1 Definiteness of symmetric matrices
12.2 Connection with quadratic forms
12.3 Sylvester’s criterion (for PD and ND)
12.4 Sylvester’s criterion (for PSD and NSD)
12.5 Applications : General second derivative tests |
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第13週 |
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13.1 Introduction of Markov chain
13.2 Convergence of a matrix power
13.3 Gershgorin’s Disk Theorem
13.4 Perron-Frobenius Theorem (I) |
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第14週 |
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14.1 Regular stochastic matrices
14.2 Perron-Frobenius Theorem (II)
14.3 Google PageRank algorithm
14.4 Approximating the PageRank vector |
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第15週 |
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15.1 Groups : Definition and Examples
15.2 Group actions
15.3 Representations
15.4 G-invariant subspaces, irreducible representations
15.5 Characters
15.6 Tensor products, symmetric and exterior powers
15.7 Character table |
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第16週 |
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Final Exam Week |
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