Coordinate System Hungyi Lee Outline Coordinate Systems Each

Coordinate System Hung-yi Lee

Outline • Coordinate Systems • Each coordinate system is a “viewpoint” for vector representation. • The same vector is represented differently in different coordinate systems. • Different vectors can have the same representation in different coordinate systems. • Changing Coordinates • Reference: textbook Ch 4. 4

Coordinate System

Vector {e 1, e 2} is a coordinate system New Coordinate System

Vector drink home

Vector

Vector home

Coordinate System • Every vector should have representation Unique representation

Why Basis? • Unique? B is independent a 1 b 1 = a 2 b 2 = = ak bk = 0

Coordinate System •

Coordinate System vector (standard vectors) (直角坐標系)

Other System → Cartesian

Other System → Cartesian • (matrix-vector product)

Cartesian → Other System find [v]B B is invertible (? ) independent

• Let B = {b 1 , b 2 , , bn} = c 1 b 1 + c 2 b 2 + + cn b n (Standard vector)

Changing Coordinates

Equation of ellipse Rotate 45◦ 2 3 ?

Equation of ellipse Use another coordinate system 2 3 What is the equation of the ellipse in the new coordinate system?

Equation of ellipse

Equation of hyperbola 30◦

Equation of hyperbola 30◦

Equation of hyperbola

Summary vector (standard vectors)