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

2D Geometry: Points

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

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

2D Geometry and Linear Algebra Points Cartesian Coordinates Vectors

2D Geometry: Distance

2D Geometry: Distance How to express distance algebraically using coordinates???

Algebra: Vector Operations Vector Addition Scalar Multiplication

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

Geometry of Vector Operations

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

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

Norm: Distance to the Origin Norm of a vector:

Distance of Between Two Points v w v - w

Dot-Product (Inner Product) and Norm

Angle Between Two Vectors

Polar Coordinate r v

Dot Product: Angle and Length Cosine Formula v w

Perpendicular Vectors v is perpendicular to w if and only if

Vector Inequalities Triangle Inequality Schwarz Inequality Proof:

3D Points z y x

3D Vector y x z

Row and Column Representation

Algebra: Vector Operations Vector Addition Scalar Multiplication

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}

Geometry of Linear Combination u v u

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

Dot-Product (Inner Product) and Norm

Vector Inequalities Triangle Inequality Schwarz Inequality Proof:

Dimensions One Dimensional Geometry Two Dimensional Geometry Three Dimensional Geometry High Dimensional Geometry

n-Dimensional Vectors and Points Transpose of vectors

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

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}

High Dimensional Algebra and Geometry Triangle Inequality Schwarz Inequality

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

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