1. Linear Algebra: What are we going to learn? 李宏毅 Hung-yi Lee

2. What are we going to learn? A system of linear equations ( 多元一次聯立方程式 ) A linear equation a system of linear equations variables coefficients constant term I believe you know how to solve it.

3. What are we going to learn? A system of linear equations ( 多元一次聯立方程式 ) m equations n variables In this course, m and n can be large

4. What are we going to learn? Solving a system of linear equations Does it have solution? Does it have unique solution? How to find the solution? Different view from high school Determinants ( 行列式 ) Beyond 3 X 3 Approximate Solution span Independent rank

5. Linear System We use a system of linear equations to describe a linear system What is system? function, transformation, operator A system has input and output E.g. Speech Recognition System: The input is audio signals, and the output is recognition results. E.g. Siri: You say “Hi, Siri”, it reply “What can I do for you?” E.g. Communication system Can have multiple inputs and outputs System x1x1 x2x2 y1y1 y2y2

6. Linear System Linear system have two properties 1. Persevering Multiplication 2. Persevering Addition Linear System xy Linear System kxky x2x2 y2y2 x 1 +x 2 y 1 +y 2 Linear System x1x1 y1y1 Linear System Linear System

7. Linear System Linear system have two properties 1. Persevering Multiplication 2. Persevering Addition System xx2x2 Linear System x Linear System kx x2x2 k2x2k2x2 Linear System Linear System Linear System

8. Linear System Two properties 1. Persevering Multiplication 2. Persevering Addition System

9. Linear System v.s. System of Linear Equations Linear System ? ……

10. Linear System …… Linear System …… Linear System …… Linear System …… A linear system is described by a system of linear equations

11. Applications + - input output ( 大二上 ) Most system are (assumed to) be linear Circuit Linear Input: voltage source, current source output: voltage and current on the load ( 燈泡、引擎 ) Linear System

12. Applications ( 大二下 ) Most system are (assumed to) be linear Linear Transmitter Receiver Communication System Linear System Channel

13. Applications Machine Learning E.g. Prediction 股票 分析 坪數 氣象 預報 房價 分析 位置 Linear system 價格 ……. 今天天氣 昨天天氣 ……. 明天天氣 今天行情 明天行情 Linear system

14. Applications We will learn EigenXXXXX Filter Linear System …… Input equals to outputOutput is zero Linear System ……

15. Applications – Image Compression 壓縮比 : 25 壓縮比 : 12.5 壓縮比 : 5 壓縮比 : 2.5 原圖

16. Linear Algebra is Important