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

2. 2D Geometry: Points

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

4. 2D Linear Algebra: Vectorsy
(a,b) x

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

6. 2D Geometry: Distance

7. 2D Geometry: Distance
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

11. {cv + d w : c, d are real numbers}
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 Pointsv w
v - w

15. Dot-Product (Inner Product) and Norm

16. Angle Between Two Vectors

17. Polar Coordinate r v

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 z y x

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 Combinationu v u

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 v1 (line)
{c v1 : c is a real number}
Linear combination of v1 and v2 (plane)
{c1 v1 + c2 v2 : c1 ,c2 are real numbers}
Linear combination of d vectors v1 , v2 ,…, vd
(d Space)
{c1v1 +c2v2+…+ cdvd : c1,c2 ,…,cd 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)