Applied Maths- Linear Algebra, Vector Analysis - 106 Academic Year
應用數學 - 線性代數 - 106學年度
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。
課程用書：
| 週次 | 課程內容 | 課程影音 | 課程下載 |
|---|---|---|---|
| 第一週 | Cartesian Plane、Vector in The Cartesian Plane、Vector Addition | 線上觀看 | MP4下載 |
| Vector Addition、Vector in R^3 and R^n、Algebraic Properties of Vector in R^n and Proof | 線上觀看 | MP4下載 | |
| Using Algebraic Properties for computation、Independent Vector Spanning R^2 Space、Linear Combination、Binary Vector | 線上觀看 | MP4下載 | |
| Dot Product、Definition and Example、Algebraic Properties of Dot Product | 線上觀看 | MP4下載 | |
| Length、Unit Vector、The Cauchy-Schwartz and Triangle Inequality、Distance between Two Vector | 線上觀看 | MP4下載 | |
| Angle between Two Vectors、Angle and Orthogonal Property、Dot Operation and Orthogonality of Function、Pythagoras’ Theorem and Projection | 線上觀看 | MP4下載 | |
| Lines in R^2 and R^3 | 線上觀看 | MP4下載 | |
| Lines in R^2 and R^3、Planes in R^3 | 線上觀看 | MP4下載 | |
| Distance from a Point to a Line/Plane、Applications | 線上觀看 | MP4下載 | |
| 第二週 | Linear or Nonlinear Equation、A system of Linear Equation in R^2 and R^3、Coefficient and Augmented Metrices | 線上觀看 | MP4下載 |
| Row Echelon Form、Elementary Row Operation and Row Equivalent、Gaussian Elimination | 線上觀看 | MP4下載 | |
| Rank and Gaussian-Jordan Elimination、Gaussian-Jordan Elimination、Homogeneous Systems、Linear Systems over Z_p | 線上觀看 | MP4下載 | |
| Vector Concepts for Linear Combinations、Spanning Set for Space R^n | 線上觀看 | MP4下載 | |
| Spanning Set for Space R^n、Linear independence | 線上觀看 | MP4下載 | |
| Linear independence | 線上觀看 | MP4下載 | |
| Application(Chemical、Network Analysis) | 線上觀看 | MP4下載 | |
| Application(Circuit、Economic Models) | 線上觀看 | MP4下載 | |
| Application(Finite Linear Games)、Numerical Solutions-Iterative Methods | 線上觀看 | MP4下載 | |
| 第三週 | Introduction to Matrix Operations、Matrix Operations、Matrix’s Equality | 線上觀看 | MP4下載 |
| Matrix Addition and Scalar Multiplication、Matrix Multiplication、Partitioned Matrices | 線上觀看 | MP4下載 | |
| Partitioned Matrices、Matrix Power、The Transpose of a Matrix | 線上觀看 | MP4下載 | |
| Properties of Addition and Scalar Multiplication、Properties of Matrix Multiplication | 線上觀看 | MP4下載 | |
| Properties of The Transpose、The inverse of a Matrix | 線上觀看 | MP4下載 | |
| Properties of Invertible Matrices、Elementary Matrices | 線上觀看 | MP4下載 | |
| The Fundamental Theorem of Invertible Matrices、The Gaussian-Jordan Method for Computing The Inverse | 線上觀看 | MP4下載 | |
| The Gaussian-Jordan Method for Computing The Inverse、The LU Factorization | 線上觀看 | MP4下載 | |
| An Easy Way to Find LU Factorizations、The P^TLU Factorization、Computational Consideration | 線上觀看 | MP4下載 | |
| 第四週 | Subspace、Row and Column Space of Matrice(Introduction) | 線上觀看 | MP4下載 |
| Row and Column Space of Matrice、Basis and Effective Procedures to Find Basis | 線上觀看 | MP4下載 | |
| Dimension、Rank、The Fundamental Theorem of Invertible Matrices | 線上觀看 | MP4下載 | |
| Introduction to Linear Transformations、Linear Transformations | 線上觀看 | MP4下載 | |
| Linear Transformations、Rotation About The Origin、Composition of Two Transformations | 線上觀看 | MP4下載 | |
| Composition of Two Transformations、Inverse of Linear Transformations、Associativity | 線上觀看 | MP4下載 | |
| Introduction to Markov Chain、Markov Chain-State Vector、Transition Matrix and Steady State Vector | 線上觀看 | MP4下載 | |
| Application Example of Markov Chain、Linear Economic Model-Leontief Model、Productive、Population Growth-Leslie Model | 線上觀看 | MP4下載 | |
| Population Growth-Leslie Model、Graph、Digraph | 線上觀看 | MP4下載 | |
| Digraph、Error-Correction Codes | 線上觀看 | MP4下載 | |
| 第五週 | Introduction to Eigenvalues and Eigenvectors、Eigenspace | 線上觀看 | MP4下載 |
| Eigenspace、Eigenpicture | 線上觀看 | MP4下載 | |
| Eigenpicture-Zero Determinant、Calculate Eigenvalues | 線上觀看 | MP4下載 | |
| Concept of Determinant and How to calculate、Determinant of n×n matrices、The Laplace Expansion | 線上觀看 | MP4下載 | |
| The Examples of Laplace Expansion、Properties of Determinants | 線上觀看 | MP4下載 | |
| Determinants of Elementary Matrices、Determinants and Matrix Operations、Cramer’s Rule and The Adjoint | 線上觀看 | MP4下載 | |
| Eigenvalues of a Square Matrix | 線上觀看 | MP4下載 | |
| Eigenvalues of a Square Matrix、The Fundamental Theorem of Invertible Matrices | 線上觀看 | MP4下載 | |
| Polynomial of a matrix A and Its corresponding eigenvalues | 線上觀看 | MP4下載 | |
| 第七週 | Similar Matrices | 線上觀看 | MP4下載 |
| Diagonalization | 線上觀看 | MP4下載 | |
| Diagonalization、The Diagonalization Theorem | 線上觀看 | MP4下載 | |
| The Power Method | 線上觀看 | MP4下載 | |
| The Power Method、The Shifted Power Method、The Inverse Power Method | 線上觀看 | MP4下載 | |
| The Shifted Inverse Power Method、Gerschgorin’s Theorem | 線上觀看 | MP4下載 | |
| Markov Chains | 線上觀看 | MP4下載 | |
| Population Growth、Leslie matrix、Perron-Frobenius Theorem | 線上觀看 | MP4下載 | |
| Linear Recurrence Relation | 線上觀看 | MP4下載 | |
| System of Linear Differential Equations | 線上觀看 | MP4下載 | |
| Exponential of A - e^A、Discrete Linear Dynamical Systems | 線上觀看 | MP4下載 | |
| Discrete Linear Dynamical Systems | 線上觀看 | MP4下載 | |
| 第八週 | Orthogonal Sets of Vectors、Coordinate | 線上觀看 | MP4下載 |
| Orthogonal Matrices、Orthogonal Complements | 線上觀看 | MP4下載 | |
| Orthogonal Projections、Orthogonal Decompositions | 線上觀看 | MP4下載 | |
| The Gram-Schmidt Process、QR Factorization | 線上觀看 | MP4下載 | |
| Orthogonal Diagonalizable | 線上觀看 | MP4下載 | |
| Spectral Theorem | 線上觀看 | MP4下載 | |
| Dual Codes-The generated Code Space C、Dual Codes-The Same or Equivalent Code | 線上觀看 | MP4下載 | |
| Quadratic Forms、Quadratic Forms-diagonalization and Principle Axes Theorem | 線上觀看 | MP4下載 | |
| Quadratic Forms、Definition and Example、Diagonalization and Principal Axes Theorem | 線上觀看 | MP4下載 | |
| The Fundamental Theorem of Quadratic Forms | 線上觀看 | MP4下載 | |
| Graphing Quadratic Equations、Identify and graph the conic | 線上觀看 | MP4下載 | |
| Graphing Quadratic Equations、Identify the quadric surface of the equation | 線上觀看 | MP4下載 | |
| 第九週 | Vector Space、Examples of Kinds of Vector Space | 線上觀看 | MP4下載 |
| Examples of Vector or Non-Vector Space、Zero and Negative vector、Subspace | 線上觀看 | MP4下載 | |
| Spanning Set | 線上觀看 | MP4下載 | |
| Linear Independence、 Bases | 線上觀看 | MP4下載 | |
| Coordinate | 線上觀看 | MP4下載 | |
| Dimension | 線上觀看 | MP4下載 | |
| Change-of-Basis Matrices | 線上觀看 | MP4下載 | |
| The Gauss-Jordan Method for Computing a change-of-Basis Matrix | 線上觀看 | MP4下載 | |
| 第十週 | Linear Transformations、Definition and Example | 線上觀看 | MP4下載 |
| Properties of Linear Transformations,、Composition of Linear Transformations, Inverse of Linear Transformation | 線上觀看 | MP4下載 | |
| The Kernel and Range of a Linear Transformations、Definition and Example | 線上觀看 | MP4下載 | |
| The Kernel and Range of a Linear Transformations、The Rank Theorem、One-to-One and Onto Linear Transformations | 線上觀看 | MP4下載 | |
| The Kernel and Range of a Linear Transformations、The Rank Theorem、One-to-One and Onto Linear Transformations | 線上觀看 | MP4下載 | |
| Linear Transformations Represented by a Matrix | 線上觀看 | MP4下載 | |
| Matrices of Composite and Inverse Linear Transformations | 線上觀看 | MP4下載 | |
| Change of Basis and Similarity | 線上觀看 | MP4下載 | |
| The Fundamental Theorem of Invertible Matrices | 線上觀看 | MP4下載 | |
| 1st Order Homogeneous,Linear,Differential,Equations、2nd Order Homogeneous,Linear,Differential,Equations | 線上觀看 | MP4下載 | |
| 第十一週 | Extends from 1D to 3D Vector Space、Scalar Functions , Functions of Scalar of Functions of Vector、Vector Functions of Vector | 線上觀看 | MP4下載 |
| Vector-Function Curve,3D to 1D Vsinf Parameter 、Find the Intersection Curve 、Limits of a Vector Functions 、Continuity and Derivative of Vector Function | 線上觀看 | MP4下載 | |
| Calculate the tangent line of a curve、Differentiation and Integration of the Curve、 The length of a Curve、The Arc length Parameter | 線上觀看 | MP4下載 | |
| Trajectory of a particle | 線上觀看 | MP4下載 | |
| Calculation of Curvature | 線上觀看 | MP4下載 | |
| Normal and Binormal Vectors、Tangent, Normal and Binormal Vectors、Another Method for the Calculation of Curvature、Curvature of Twisted Cubic | 線上觀看 | MP4下載 | |
| Level Curves for Functions of two variables、Functions of three variables F(x,y,z)、 Partial Derivatives of Functions | 線上觀看 | MP4下載 | |
| Scalar Functions to Vector Functions、Gradient Calculation | 線上觀看 | MP4下載 | |
| Concept of Gradient Calculation、Gradient Calculation | 線上觀看 | MP4下載 | |
| 第十三週 | Level Curves and Gradient | 線上觀看 | MP4下載 |
| Vector fields,The del operator | 線上觀看 | MP4下載 | |
| Concept of Divergence Calculation、Curl less or Divergent less Potential、Scalar or Vector Potential | 線上觀看 | MP4下載 | |
| Line Integrals on a 2D plane | 線上觀看 | MP4下載 | |
| Line Integrals on a 2D plane | 線上觀看 | MP4下載 | |
| Line Integrals on a 2D plane-Plane Work done by force | 線上觀看 | MP4下載 | |
| Path Independence Integration Result、Conservative Vector Fields | 線上觀看 | MP4下載 | |
| Conservative Vector Fields | 線上觀看 | MP4下載 | |
| Test for a Conservative Vector field | 線上觀看 | MP4下載 | |
| 第十四週 | Introduction to Double Integrals、Applications of Double Integrals | 線上觀看 | MP4下載 |
| Calculation of Double Integrals、Change of Variables、Double Integrals in Poar coordinates | 線上觀看 | MP4下載 | |
| Change of variables | 線上觀看 | MP4下載 | |
| Proof of Green’s Theorem | 線上觀看 | MP4下載 | |
| Green’s Theorem and Calculation 、Stoke’s Theorem、Application of Green’s Theorem | 線上觀看 | MP4下載 | |
| Application of Green’s Theorem、Apply Green’s Theorem in the region with holes | 線上觀看 | MP4下載 |
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
課程專區
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