1. Lecture 2: Geometry vs Linear Algebra Points-Vectors and Distance-Norm Shang-Hua Teng

2. 2D Geometry: Points

3. 2D Geometry: Cartesian Coordinates x y (a,b)

4. 2D Linear Algebra: Vectors x y (a,b) 0

5. 2D Geometry and Linear Algebra Points Cartesian Coordinates Vectors

6. 2D Geometry: Distance

7. How to express distance algebraically using coordinates???

8. Algebra: Vector Operations Vector Addition Scalar Multiplication

9. Geometry of Vector Operations Vector Addition: v + w v w v + w

10. Geometry of Vector Operations -v v 2v

11. Linear Combination Linear combination of v and w {cv + d w : c, d are real numbers}

12. Geometry of Vector Operations Vector Subtraction: v - w v w v + w v - w

13. Norm: Distance to the Origin Norm of a vector:

14. Distance of Between Two Points v w v - w

15. Dot-Product (Inner Product) and Norm

16. Angle Between Two Vectors v w

17. Polar Coordinate v r

18. Dot Product: Angle and Length Cosine Formula v w

19. Perpendicular Vectors v is perpendicular to w if and only if

20. Vector Inequalities Triangle Inequality Schwarz Inequality Proof:

21. 3D Points y x z

22. 3D Vector y x z

23. Row and Column Representation

24. Algebra: Vector Operations Vector Addition Scalar Multiplication

25. Linear Combination Linear combination of v (line) {cv : c is a real number} Linear combination of v and w (plane) {cv + d w : c, d are real numbers} Linear combination of u, v and w (3 Space) {bu +cv + d w : b, c, d are real numbers}

26. Geometry of Linear Combination u u v

27. Norm and Distance Norm of a vector: Distance y x z

28. Dot-Product (Inner Product) and Norm

29. Vector Inequalities Triangle Inequality Schwarz Inequality Proof:

30. Dimensions One Dimensional Geometry Two Dimensional Geometry Three Dimensional Geometry High Dimensional Geometry

31. n-Dimensional Vectors and Points Transpose of vectors

32. High Dimensional Geometry Vector Addition and Scalar Multiplication Dot-product Norm Cosine Formula

33. High Dimensional Linear Combination Linear combination of v 1 (line) {c v 1 : c is a real number} Linear combination of v 1 and v 2 (plane) {c 1 v 1 + c 2 v 2 : c 1,c 2 are real numbers} Linear combination of d vectors v 1, v 2,…, v d (d Space) {c 1 v 1 +c 2 v 2 +…+ c d v d : c 1,c 2,…,c d are real numbers}

34. High Dimensional Algebra and Geometry Triangle Inequality Schwarz Inequality

35. Basic Notations Unit vector ||v||=1 v/||v|| is a unit vector Row times a column vector = dot product

36. Basic Geometric Shapes: Circles (Spheres), Disks (Balls)