Linear Function in Coordinate System Hung-yi Lee

Outline Describing a function in a coordinate system A complex function in one coordinate system can be simple in other systems. Reference: Textbook Chapter 4.5 Why we have to do that?

Basic Idea Simple Function Another coordinate system output’ Input’ Why we have to do that? Cartesian coordinate system Input Output Complex Function

Sometimes a function can be complex …… Example: reflection about a line L through the origin in R2 𝑥 1 𝑥 2 𝑇 𝑥 1 𝑥 2 =? T = 𝑇 𝑒 1 𝑇 𝑒 2 =? 𝑇 𝑥 1 𝑥 2

Sometimes a function can be complex …… Example: reflection about a line L through the origin in R2 special case: L is the horizontal axis 𝑇′ 𝑥 1 𝑥 2 =? 𝑥 1 −𝑥 2 𝑥 1 𝑥 2 A similar but simpler function 𝑇′ = 1 0 0 −1 𝑇′ 𝑥 1 𝑥 2 𝑇′ 𝑒 1 𝑇′ 𝑒 2 = 𝑒 1 =− 𝑒 2

Describing the function in another coordinate system Example: reflection about a line L through the origin in R2 In another coordinate system B …. B = 𝑏 1 , 𝑏 2 𝑇 𝑏 2 =− 𝑏 2 𝑏 2 𝑏 1 𝑇 𝑏 1 = 𝑏 1

Describing the function in another coordinate system Example: reflection about a line L through the origin in R2 = 1 0 0 −1 𝑇 B In another coordinate system B …. Input and output are both in B 𝑇 𝑏 1 = 𝑏 1 𝑏 2 B = 𝑒 2 T 下標 B 上標 gamma 𝑇 B 𝑏 1 B = 𝑏 1 B 𝑇 B 𝑒 1 = 𝑒 1 𝑇 𝑏 2 =− 𝑏 2 𝑏 1 B = 𝑒 1 𝑇 B 𝑏 2 B = − 𝑏 2 B 𝑇 B 𝑒 2 =− 𝑒 2

Flowchart 𝑣 B 𝑇 𝑣 B 𝑣 𝑇 𝑣 = 1 0 0 −1 𝑇 B 𝑇 =? (B matrix of T) 𝑏 1 𝑏 2 = 1 0 0 −1 𝑇 B (B matrix of T) 𝑣 B 𝑇 𝑣 B 𝑏 1 𝑏 2 reflection about the horizontal line 𝑒 1 𝑒 2 𝑇 =? 𝑣 𝑇 𝑣 reflection about a line L

𝑣 B 𝑇 𝑣 B 𝐵 𝐵 −1 𝑣 𝑇 𝑣 𝐴=𝐵 𝑇 B 𝐵 −1 𝑇 B = 𝐵 −1 A𝐵 Flowchart 𝑇 B 𝑇 =? B coordinate system 𝐵 𝐵 −1 Cartesian coordinate system 𝑇 =? 𝑣 𝑇 𝑣 𝐴=𝐵 𝑇 B 𝐵 −1 𝑇 B = 𝐵 −1 A𝐵 similar similar

Example: reflection operator T about the line y = (1/2)x 𝐵 −1 = 0.4 0.2 −0.2 0.4 𝐵= 2 −1 1 2 𝑏 2 = −1 2 y = (1/2)x 𝑇 B 𝑣 B 𝑇 𝑣 B 𝑏 1 = 2 1 1 0 0 −1 𝐵 −1 𝐵 y = (1/2)x 𝑒 2 = 0 1 𝐴=? 𝑣 𝑇 𝑣 𝑒 1 = 1 0 𝐴=𝐵 𝑇 B 𝐵 −1

Example: reflection operator T about the line y = (1/2)x 𝐵 −1 = 0.4 0.2 −0.2 0.4 𝐵= 2 −1 1 2 0.6 0.8 0.8 −0.6 𝐴= 𝑇 B 𝑣 B 𝑇 𝑣 B y = (1/2)x 𝑐 2 = −0.5 1 1 0 0 −1 𝐵 −1 𝑐 1 = 4 2 𝐵 0.6000 0.8000 0.8000 -0.6000 𝐴=? 𝐴=𝐶 𝑇 C 𝐶 −1 𝑣 𝑇 𝑣 = 0.6 0.8 0.8 −0.6 𝐴=𝐵 𝑇 B 𝐵 −1

Example 2 (P279) 𝑇 B =? 𝑣 B 𝐴= 3 0 1 1 1 0 −1 −1 3 𝑇 𝑣 B 𝑇 B = 𝐵 −1 𝐴𝐵 𝐴= 3 0 1 1 1 0 −1 −1 3 𝑇 𝑣 B 𝑇 B = 𝐵 −1 𝐴𝐵 𝐵= 1 1 2 1 2 1 1 3 1 𝐵 −1 𝐵 3 -9 8 -1 3 -3 1 6 1 𝐴 is known 3 −9 8 −1 3 −3 1 6 1 𝑣 𝑇 B = 𝑇 𝑣 𝐴=𝐵 𝑇 B 𝐵 −1

Example 3 (P279) Determine T b1 c1 b2 c2 b3 c3 e1 e2 e3 𝑣 B 𝑇 𝑣 B b1, b2, b3 as a coordinate system 𝐵 −1 𝐵 −1 {b1, b2, b3} is a basis of R3 Simple ….. A = C B ^ -1 𝐴 b1 b2 b3 𝑣 𝑇 𝑣 c1 c2 c3

Example 3 (P279) Determine T 𝑇 B = 𝐵 −1 𝑐 1 𝐵 −1 𝑐 2 𝐵 −1 𝑐 3 = 𝐵 −1 𝐶 = 𝐵𝐵 −1 𝐶 𝐵 −1 =𝐶 𝐵 −1 e1 e2 e3 𝑣 B 𝑇 𝑣 B B-1c1 B-1c2 B-1c3 𝑇 B b1, b2, b3 as a coordinate system 𝐵 −1 𝐵 −1 𝐵 {b1, b2, b3} is a basis of R3 𝐴 b1 b2 b3 𝑣 𝑇 𝑣 c1 c2 c3

Cartesian coordinate system Inception 小開的父親說：  "I'm disappointed that you're trying so hard to be me." 小開有了不要繼承父業的念頭 𝑇 B B coordinate system 𝑣 B 𝑇 𝑣 B 𝑇 B = 𝐵 −1 𝐴𝐵 夢境 𝐵 清醒 Cartesian coordinate system 做夢 𝐵 −1 𝑇 𝑣 𝑇 𝑣 現實 𝑇 =𝐵 𝑇 B 𝐵 −1 小開解散公司 說服小開解散公司