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Preface

Preface. Linear Algebra Ming-Feng Yeh Department of Electrical Engineering Lunghwa University of Science and Technology. Introduction.

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Preface

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  1. Preface Linear Algebra Ming-Feng Yeh Department of Electrical Engineering Lunghwa University of Science and Technology

  2. Introduction • Elementary Linear Algebra is designed for the introductory linear algebra generally taken by sophomores and juniors majoring in engineering, computer science, mathematics, economics, statistic, or operations research. • The primary prerequisite for this course isalgebra, but we also assume familiarity with analytic geometry and trigonometry. • Calculus is not a prerequisite for this course, but some examples and exercises require a knowledge of calculus. Preface

  3. What is Linear Algebra? • The most fundamental theme of linear algebra, and the first topic covered in this textbook, is the theory of systems of linear equations. • Let x represent the speed of theplane and y the speed of the wind,then the following system modelsthe problem. Preface

  4. Systems of Linear Equations • The system of two equations and two unknowns.The solution is x = 900 kilometers per hour and y = 100 kilometers per hour. • Geometrically, this systemrepresents two lines in thexy-plane. Two lines intersectat the point (900, 100), whichverifies the answer we obtained. y x + y = 1000 (900, 100) x xy = 900 Preface

  5. Topics of Linear Algebra • Solving systems of linear equations is one of the most important applications of linear algebra. • Topics of linear algebra: • The fundamental topics – linear systems, matrices and determinants • The central theoretical topic – vector space • Another major focus – engenvalue and eigenvector Preface

  6. 高中（職）數學回顧 #1 直線方程式之求法： • 兩點式： • 點斜式：（已知一點及其斜率）已知之斜率為已知線上之一點為 (x0, y0)則直線方程式為 y = mx + b其中y0 = mx0 + b y = mx + b y (x2, y2) y (x1, y1) x x (x0, y0) Preface

  7. 高中（職）數學回顧 #2 • 考慮 xy平面(xy – plane)上之任意兩直線： • 若 即 此時兩直線不平行,不平行之兩直線必交於一點。 • 此線性方程式系統在此情況之下只有唯一解。 • 定義 若 時, 表示也就是說, 兩直線在 時會交於一點。 • 『 』為行列式值，將在第3章進一步介紹。 Preface

  8. 高中（職）數學回顧 #3 • 若 即 或 此時兩直線可能平行或重合。 • 若 表兩直線重合，此時有無窮多組解。 • 若 表兩直線平行，此時有無解。 Preface

  9. 高中（職）數學回顧 #4 • 試求出以下線性方程式系統之解： 1. 將第(1)式與第(2)式對調，不影響 “解” 之結果。 2. 將第(3)式乘以(–2)加至第(4)式，可得 3.將第(5)式乘以(–1)可得 4.將第(6)式代回第(3)式，可得 （兩線交於一點） Preface

  10. 高中（職）數學回顧 #5 • 試求出以下線性方程式系統之解： 將第(1)式乘以(–2)加至第(2)式，可得 可發現第(3)式為一矛盾方程式， 故此線性方程式系統無解。 （兩平行直線） Preface

  11. 高中（職）數學回顧 #6 • 試求出以下線性方程式系統之解： 將第(1)式乘以(–2)加至第(2)式，可得 可發現第(3)式為一恆等式， 故此線性方程式系統無窮多解。（兩平行重合） 令 y = t, t R, 則 x = 4 – 2t.則此線性方程式系統之解集合為：{(4 – 2t, t): t R} Preface

  12. y L1 (x1, y1) (x2, y2) (x0, y0) x L2 高中（職）數學回顧 #7 垂直之兩直線 • 特例：兩座標軸 x = 0 (m1 = 0)和 y = 0 (m2 = )必為垂直之兩直線 • 通式： • 若 則當 時，L1與 L2垂直。 Preface

  13. y (x1, y1) (x2, y2) (0, 1) L1 L2 x (1, 0) 高中（職）數學回顧 #8 • 向量為一有方向性之線段（有方向、大小之量，起點為原點） • 圖示之兩向量為(x1, y1) 和 (x2, y2) • 內積之定義： • 當內積等於0時，表兩向量垂直。即 • 例：(x1, y1) = (1, 0), (x2, y2) = (0, 1)，則 Preface

  14. 94數學考科 • 兩射線OA與OB交於O點，試問下列選項中哪些向量的終點會落在陰影區域內？   Preface

  15. 94數學考科 O • 座標平面上一鳶形ABCD，其中A,C在y-軸，B,D在x-軸上，且 ， ， 。令 分別表直線AB, BC, CD, DA之斜率。試問下列哪些敘述成立？    Preface

  16. 94數學考科 • 假設座標空間中三相異平面E1、 E2、 E3皆通過(1,2,0)與(3,0,2)兩點，試問以下哪些點也同時在此三平面上？(1) (2,2,2); (2) (1,1,1); (3) (4, 2,2); (4) (2,4,0);(5) (5, 4, 2). 解：E1、 E2、 E3三平面共線，所求之點 P若落在通過A(1,2,0)與B(3,0,2)兩點之直線上，即同時在此三平面上。三點共線問題 Preface

  17. P  A B Q 94數學考科 • ABCD-EFGH為邊長等於1之正立方體。若P 點在立方體之內部且滿足 ，則P點至直線AB之距離為何？ 解：由題意可知， Preface

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