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課程名稱 |
線性代數導論一
Introduction to Linear Algebra Ⅰ(Ⅰ) |
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開課學期 |
112-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
蔡國榮 |
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課號 |
MATH4018 |
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課程識別碼 |
201E49950 |
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班次 |
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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 |
新103新103 |
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備註 |
本課程以英語授課。數學系學生不計入畢業學分。輔系生修習得替代線性代數一。
限外系（所）學生
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 |
Many problems in science are often resolved by firstly linearizing them. For instance, in calculus, differentiation can be seen as a process of locally approximating functions with linear functions. This process of linearization often sheds new light into the original problem. Linear algebra, the branch of mathematics that systematically deals with linearized functions, is now considered the fundamental language in almost all quantitative scientific studies.
The objective of this course is to introduce the fundamental ideas of linear algebra. In the first semester, our focus will be on studying vector spaces and the linear maps defined between them. Although these concepts are defined in abstract and general terms, we will approach them in a concrete and computable manner using matrices. By the end of the first semester, we will discover that every linear map on a finite dimensional vector space can be represented by an upper triangular matrix, and in some cases, even better, by a diagonal matrix, depending on the choice of a `basis’.
This course primarily focuses on pure mathematics. Theorems will be rigorously proven in their most general form, but key concepts and results will be illustrated through concrete examples from various disciplines.
Syllabus 預定每週教學進度
(Part I. Vocabulary in Linear Algebra)
Week 1. Fields and Vector spaces, Subspaces
Week 2. Linear independence, Spanning, Basis and Dimension
Week 3. Direct sums and complementary subspaces
Week 4. Linear maps (I) : kernel and image, rank and nullity theorem
Week 5. Linear maps (II) : representation by matrices
Week 6. Linear maps (III) : Gaussian elimination, rank of a matrix
Week 7. Linear maps (IV) : inverse of a linear map, transition matrices
Week 8. Midterm Exam Week
(Part II. Theory of Diagonalization)
Week 9. Determinants (I) : Definitions, properties, Cramer's rule
Week 10. Determinants (II) : Multi-linearity, Block matrices
Week 11. Diagonalization (I) : Eigenvalues, eigenvectors and eigenspaces
Week 12. Diagonalization (II) : algebraic and geometric multiplicities
Week 13. Diagonalization (III) : Cayley-Hamilton, minimal polynomials
Week 14. Applications to differential and difference equations
Week 15. Dual spaces and duality theorems
Week 16. Final Exam Week |
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課程目標 |
After finishing this course, students are expected to
1. understand the meanings of linearly independent vectors, spanning sets and bases of vector spaces ;
2. understand the equivalence of linear maps between vector spaces and matrices ;
3. be able to row reduce a matrix, compute its rank and solve systems of linear equations ;
4. be able to define a determinant in all dimensions, together with applications and techniques for calculating determinants ;
5. be able to define the eigenvalues and eigenvectors of a linear map or matrix, and know how to calculate them ;
6. be able to define similar matrices and their algebraic properties ;
7. be able to state and prove criteria for a given matrix to be diagonalizable and the application of diagonalization in solving difference/differential equations. |
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課程要求 |
Prior experience in basic operations of matrices (e.g. addition, subtraction, multiplication) are desirable.
First year calculus courses can be taken in parallel with this course : occasionally we may employ results and examples from calculus. |
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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 follow the following book closely :
S. H. Friedberg, A. J. Insel, L. E. Spence, "Linear Algebra", 4th Edition, Pearson Education, 2014 ; ISBN, 0321998898. |
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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".
3. H. Dym, "Linear Algebra in Action". |
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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 |
以錄音輔助, 以錄影輔助, 提供學生彈性出席課程方式 |
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作業繳交方式 |
學生與授課老師協議改以其他形式呈現 |
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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 Motivation of abstract linear algebra
1.2 Fields
1.3 Vector spaces
1.4 Basic properties of vector spaces
1.5 Subspaces |
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第2週 |
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2.1 Motivation
2.2 Linear dependence and independence
2.3 Spanning sets
2.4 Basis
2.5 Dimension |
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第3週 |
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3.1 Dimensions of standard spaces
3.2 Consequence of Replacement Theorem
3.3 Sifting and proof of Replacement Theorem
3.4 Direct sum of subspaces |
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第4週 |
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4.1 Motivation : Matrices as functions
4.2 Linear transformations : definition and examples
4.3 Properties of linear transformations
4.4 Kernel and image
4.5 Rank-Nullity Theorem |
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第5週 |
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5.1 Isomorphism of vector spaces
5.2 Representations of linear maps as matrices
5.3 Composition of linear maps and matrix multiplications |
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第6週 |
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6.1 Inverse of a matrix
6.2 Row operations and elementary row matrices
6.3 Column operations : a parallel story
6.4 Smith Normal Form |
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第7週 |
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7.1 Recap on elementary matrices
7.2 Augmented matrices
7.3 Applications : Finding inverse of a matrix
7.4 Applications : System of linear equations
7.5 TFAE Theorem
7.6 Change of coordinate matrix |
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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 Definition of determinants
9.2 Row operations on determinants
9.3 Properties of determinants
9.4 General Laplace expansion
9.5 Adjugate matrix and inverse of a matrix
9.6 Cramer’s rule |
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第10週 |
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10.1 Determinant of a linear operator
10.2 Block matrices : multiplication
10.3 Block matrices : block inversion formula
10.4 Block matrices : determinant, Schur's factorisation
10.5 Unfinished business : Proof of Theorem 9.2.1 |
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第11週 |
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11.1 Similar matrices
11.2 Definition of diagonalizability
11.3 Eigenvalues and eigenvectors : Definition
11.4 Characteristic polynomials
11.5 Diagonalizability criterion I : Eigen-basis |
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第12週 |
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12.1 Splitting of polynomials
12.2 Algebraic and geometric multiplicities
12.3 Diagonalizability criterion II : a(λ) = g(λ)
12.4 Applications : Finding A^n (Method 1) |
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第13週 |
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13.1 Evaluating polynomials at matrices
13.2 Cayley-Hamilton Theorem
13.3 Applications : Finding A^n (Method 2)
13.4 Minimal polynomial
13.5 Diagonalizability criterion III : Minimal polynomial
13.6 Applications of Diagonalization Theorem
13.7 Proof of Cayley-Hamilton Theorem |
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第14週 |
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14.1 Spectrum Mapping Theorem
14.2 Simultaneous diagonalizability
14.3 Schur Triangulation Theorem
14.4 Systems of differential equations
14.5 Systems of difference equations |
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第15週 |
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15.1 Space of linear maps
15.2 Dual space
15.3 Dual basis
15.4 Dual linear maps and transposes
15.5 Annihilator and duality theorems
15.6 Double duals |
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第16週 |
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Final Exam Week |
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