Applied Maths- Linear Algebra, Vector Analysis - 106 Academic Year
本課程是由 國立陽明交通大學電子物理系提供。
本課程介紹在大學第二年如電磁學等專業課程所需之數學工具，包括線性代數與向量微積分，課程前2/3介紹線性代數而後1/3課程內容是向量微積分，預期學習後較易跨越進入高年級的專業領域課程。
從熟悉的空間向量出發了解線性空間的特性出發，介紹多元一次良立方程組與線性空間關聯，並引進矩陣數學工具來解問題，而持續探索矩陣、反矩陣及矩陣向量空間概念，更進一步介紹矩陣對角化與正交化的運算，最後介紹廣義的向量空間概念，並把廣義向量空間的線性轉換寫成矩陣運算，學習者可了解函數亦為廣義向量，及函數與函數之間正交的特性等，是進入進階之微分方程課程所需的基礎訓練，且能進一步理解量子物理中使用矩陣運算的廣義向量空間概念。在後1/3課程中介紹空間中曲線、曲率計算與線積分計算，並介紹向量微積分中重要的觀念-梯度、散度與旋度計算，此部分為電磁學一開始會用到的數學工具。
Textbook：
- Poole, David/ Lipsett, Roger”Linear Algebra – A Modern Introduction”Cengage Learning,2014
- D. G. Zill/W. S. Wright”Advanced Engineering Mathematics"Jones & Bartlett Learning,2016
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Instructor(s) Department of Electrophysics Prof. Wen-Bin Jian Course Credits 3 Credits Academic Year 106 Academic Year Level College Students Prior Knowledge High School Math Skills Freshman Physics Course - Calculus Skills Related Resources Course Video Course Syllabus
| Week | Course Content | Course Video |
|---|---|---|
| Week 1 | Cartesian Plane、Vector in The Cartesian Plane、Vector Addition | Watch Online |
| Vector Addition、Vector in R^3 and R^n、Algebraic Properties of Vector in R^n and Proof | Watch Online | |
| Using Algebraic Properties for computation、Independent Vector Spanning R^2 Space、Linear Combination、Binary Vector | Watch Online | |
| Dot Product、Definition and Example、Algebraic Properties of Dot Product | Watch Online | |
| Length、Unit Vector、The Cauchy-Schwartz and Triangle Inequality、Distance between Two Vector | Watch Online | |
| Angle between Two Vectors、Angle and Orthogonal Property、Dot Operation and Orthogonality of Function、Pythagoras’ Theorem and Projection | Watch Online | |
| Lines in R^2 and R^3 | Watch Online | |
| Lines in R^2 and R^3、Planes in R^3 | Watch Online | |
| Distance from a Point to a Line/Plane、Applications | Watch Online | |
| Week 2 | Linear or Nonlinear Equation、A system of Linear Equation in R^2 and R^3、Coefficient and Augmented Metrices | Watch Online |
| Row Echelon Form、Elementary Row Operation and Row Equivalent、Gaussian Elimination | Watch Online | |
| Rank and Gaussian-Jordan Elimination、Gaussian-Jordan Elimination、Homogeneous Systems、Linear Systems over Z_p | Watch Online | |
| Vector Concepts for Linear Combinations、Spanning Set for Space R^n | Watch Online | |
| Spanning Set for Space R^n、Linear independence | Watch Online | |
| Linear independence | Watch Online | |
| Application(Chemical、Network Analysis) | Watch Online | |
| Application(Circuit、Economic Models) | Watch Online | |
| Application(Finite Linear Games)、Numerical Solutions-Iterative Methods | Watch Online | |
| Week 3 | Introduction to Matrix Operations、Matrix Operations、Matrix’s Equality | Watch Online |
| Matrix Addition and Scalar Multiplication、Matrix Multiplication、Partitioned Matrices | Watch Online | |
| Partitioned Matrices、Matrix Power、The Transpose of a Matrix | Watch Online | |
| Properties of Addition and Scalar Multiplication、Properties of Matrix Multiplication | Watch Online | |
| Properties of The Transpose、The inverse of a Matrix | Watch Online | |
| Properties of Invertible Matrices、Elementary Matrices | Watch Online | |
| The Fundamental Theorem of Invertible Matrices、The Gaussian-Jordan Method for Computing The Inverse | Watch Online | |
| The Gaussian-Jordan Method for Computing The Inverse、The LU Factorization | Watch Online | |
| An Easy Way to Find LU Factorizations、The P^TLU Factorization、Computational Consideration | Watch Online | |
| Week 4 | Subspace、Row and Column Space of Matrice(Introduction) | Watch Online |
| Row and Column Space of Matrice、Basis and Effective Procedures to Find Basis | Watch Online | |
| Dimension、Rank、The Fundamental Theorem of Invertible Matrices | Watch Online | |
| Introduction to Linear Transformations、Linear Transformations | Watch Online | |
| Linear Transformations、Rotation About The Origin、Composition of Two Transformations | Watch Online | |
| Composition of Two Transformations、Inverse of Linear Transformations、Associativity | Watch Online | |
| Introduction to Markov Chain、Markov Chain-State Vector、Transition Matrix and Steady State Vector | Watch Online | |
| Application Example of Markov Chain、Linear Economic Model-Leontief Model、Productive、Population Growth-Leslie Model | Watch Online | |
| Population Growth-Leslie Model、Graph、Digraph | Watch Online | |
| Digraph、Error-Correction Codes | Watch Online | |
| Week 5 | Introduction to Eigenvalues and Eigenvectors、Eigenspace | Watch Online |
| Eigenspace、Eigenpicture | Watch Online | |
| Eigenpicture-Zero Determinant、Calculate Eigenvalues | Watch Online | |
| Concept of Determinant and How to calculate、Determinant of n×n matrices、The Laplace Expansion | Watch Online | |
| The Examples of Laplace Expansion、Properties of Determinants | Watch Online | |
| Determinants of Elementary Matrices、Determinants and Matrix Operations、Cramer’s Rule and The Adjoint | Watch Online | |
| Eigenvalues of a Square Matrix | Watch Online | |
| Eigenvalues of a Square Matrix、The Fundamental Theorem of Invertible Matrices | Watch Online | |
| Polynomial of a matrix A and Its corresponding eigenvalues | Watch Online | |
| Week 7 | Similar Matrices | Watch Online |
| Diagonalization | Watch Online | |
| Diagonalization、The Diagonalization Theorem | Watch Online | |
| The Power Method | Watch Online | |
| The Power Method、The Shifted Power Method、The Inverse Power Method | Watch Online | |
| The Shifted Inverse Power Method、Gerschgorin’s Theorem | Watch Online | |
| Markov Chains | Watch Online | |
| Population Growth、Leslie matrix、Perron-Frobenius Theorem | Watch Online | |
| Linear Recurrence Relation | Watch Online | |
| System of Linear Differential Equations | Watch Online | |
| Exponential of A - e^A、Discrete Linear Dynamical Systems | Watch Online | |
| Discrete Linear Dynamical Systems | Watch Online | |
| Week 8 | Orthogonal Sets of Vectors、Coordinate | Watch Online |
| Orthogonal Matrices、Orthogonal Complements | Watch Online | |
| Orthogonal Projections、Orthogonal Decompositions | Watch Online | |
| The Gram-Schmidt Process、QR Factorization | Watch Online | |
| Orthogonal Diagonalizable | Watch Online | |
| Spectral Theorem | Watch Online | |
| Dual Codes-The generated Code Space C、Dual Codes-The Same or Equivalent Code | Watch Online | |
| Quadratic Forms、Quadratic Forms-diagonalization and Principle Axes Theorem | Watch Online | |
| Quadratic Forms、Definition and Example、Diagonalization and Principal Axes Theorem | Watch Online | |
| The Fundamental Theorem of Quadratic Forms | Watch Online | |
| Graphing Quadratic Equations、Identify and graph the conic | Watch Online | |
| Graphing Quadratic Equations、Identify the quadric surface of the equation | Watch Online | |
| Week 9 | Vector Space、Examples of Kinds of Vector Space | Watch Online |
| Examples of Vector or Non-Vector Space、Zero and Negative vector、Subspace | Watch Online | |
| Spanning Set | Watch Online | |
| Linear Independence、 Bases | Watch Online | |
| Coordinate | Watch Online | |
| Dimension | Watch Online | |
| Change-of-Basis Matrices | Watch Online | |
| The Gauss-Jordan Method for Computing a change-of-Basis Matrix | Watch Online | |
| Week 10 | Linear Transformations、Definition and Example | Watch Online |
| Properties of Linear Transformations,、Composition of Linear Transformations, Inverse of Linear Transformation | Watch Online | |
| The Kernel and Range of a Linear Transformations、Definition and Example | Watch Online | |
| The Kernel and Range of a Linear Transformations、The Rank Theorem、One-to-One and Onto Linear Transformations | Watch Online | |
| The Kernel and Range of a Linear Transformations、The Rank Theorem、One-to-One and Onto Linear Transformations | Watch Online | |
| Linear Transformations Represented by a Matrix | Watch Online | |
| Matrices of Composite and Inverse Linear Transformations | Watch Online | |
| Change of Basis and Similarity | Watch Online | |
| The Fundamental Theorem of Invertible Matrices | Watch Online | |
| 1st Order Homogeneous,Linear,Differential,Equations、2nd Order Homogeneous,Linear,Differential,Equations | Watch Online | |
| Week 11 | Extends from 1D to 3D Vector Space、Scalar Functions , Functions of Scalar of Functions of Vector、Vector Functions of Vector | Watch Online |
| Vector-Function Curve,3D to 1D Vsinf Parameter 、Find the Intersection Curve 、Limits of a Vector Functions 、Continuity and Derivative of Vector Function | Watch Online | |
| Calculate the tangent line of a curve、Differentiation and Integration of the Curve、 The length of a Curve、The Arc length Parameter | Watch Online | |
| Trajectory of a particle | Watch Online | |
| Calculation of Curvature | Watch Online | |
| Normal and Binormal Vectors、Tangent, Normal and Binormal Vectors、Another Method for the Calculation of Curvature、Curvature of Twisted Cubic | Watch Online | |
| Level Curves for Functions of two variables、Functions of three variables F(x,y,z)、 Partial Derivatives of Functions | Watch Online | |
| Scalar Functions to Vector Functions、Gradient Calculation | Watch Online | |
| Concept of Gradient Calculation、Gradient Calculation | Watch Online | |
| Week 13 | Level Curves and Gradient | Watch Online |
| Vector fields,The del operator | Watch Online | |
| Concept of Divergence Calculation、Curl less or Divergent less Potential、Scalar or Vector Potential | Watch Online | |
| Line Integrals on a 2D plane | Watch Online | |
| Line Integrals on a 2D plane | Watch Online | |
| Line Integrals on a 2D plane-Plane Work done by force | Watch Online | |
| Path Independence Integration Result、Conservative Vector Fields | Watch Online | |
| Conservative Vector Fields | Watch Online | |
| Test for a Conservative Vector field | Watch Online | |
| Week 14 | Introduction to Double Integrals、Applications of Double Integrals | Watch Online |
| Calculation of Double Integrals、Change of Variables、Double Integrals in Poar coordinates | Watch Online | |
| Change of variables | Watch Online | |
| Proof of Green’s Theorem | Watch Online | |
| Green’s Theorem and Calculation 、Stoke’s Theorem、Application of Green’s Theorem | Watch Online | |
| Application of Green’s Theorem、Apply Green’s Theorem in the region with holes | Watch Online |
課程目標
本課程介紹在大學第二年如電磁學等專業課程所需之數學工具，包括線性代數與向量微積分，課程前2/3介紹線性代數而後1/3課程內容是向量微積分，預期學習後較易跨越進入高年級的專業領域課程。
課程特色
從熟悉的空間向量出發了解線性空間的特性出發，介紹多元一次良立方程組與線性空間關聯，並引進矩陣數學工具來解問題，而持續探索矩陣、反矩陣及矩陣向量空間概念，更進一步介紹矩陣對角化與正交化的運算，最後介紹廣義的向量空間概念，並把廣義向量空間的線性轉換寫成矩陣運算，學習者可了解函數亦為廣義向量，及函數與函數之間正交的特性等，是進入進階之微分方程課程所需的基礎訓練，且能進一步理解量子物理中使用矩陣運算的廣義向量空間概念。在後1/3課程中介紹空間中曲線、曲率計算與線積分計算，並介紹向量微積分中重要的觀念-梯度、散度與旋度計算，此部分為電磁學一開始會用到的數學工具。
課程執行方式
課程參考David Poole編寫的”Linear Algebra – A Modern Introduction”及D. G. Zill與W. S. Wright編寫的”Advanced Engineering Mathematics”書中第九章內容安排，依此兩本教課書內容分別介紹線性代數及向量微積分編制投影片及製作課程影片，本課程影片內容分三段提供三階段式學習及三次學期檢測來確認學習狀況，三階段詳述如下。
第一階段：
Week01 複習熟悉的二維或三維空間向量，了解線性空間特性
Week02 介紹多元一次線性方程組及探索其與線性向量空間的關聯
Week03 矩陣、矩陣運算、反矩陣、Gauss-Jordan方法找反矩陣、因式分解
Week04 矩陣的線性空間、基底、維度、解空間、線性轉換
第二階段：
Week05 本徵值、本徵向量、行列式
Week07 相似矩陣、矩陣對角化、迭代數值法求本徵值、對角化矩陣應用
Week08 正交矩陣、正交投影、Gram-Schmidt方法將矩陣正交化、對稱矩陣正交對角化、正交矩陣應用
Week09 廣義的向量空間、線性獨立、廣義向量空間的基底與維度
第三階段：
Week10 線性轉換與其矩陣運算
Week11 向量函數、曲線、曲率、偏微分、梯度向量運算
Week13 切面語法線、旋度與散度、線積分、路徑無關
Week14 重積分、極座標、Green的理論、面積分
COURSE
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