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:

    1. Poole, David/ Lipsett, Roger”Linear Algebra – A Modern Introduction”Cengage Learning,2014
    2. D. G. Zill/W. S. Wright”Advanced Engineering Mathematics"Jones & Bartlett Learning,2016

    為求學習成效完美,請購買課本!

     

    授課教師 電子物理系 簡紋濱老師
    課程學分 3學分
    授課年度 106學年度
    授課對象 大學生
    預備知識 高中數學能力 大一物理課程之微積分計算能力
    課程提供 課程影音   課程綱要 

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