1. Linear System = System of Linear Equations
李宏毅
Hung-yi Lee
Review system
Linear system is a system of linear equation

2. Review A system of linear equations (多元一次聯立方程式)
coefficients
variables
constant term
A linear equation
2 𝑥 𝑥 𝑥 𝑥 𝑥 2 −1 𝑥 3 −2 𝑥 𝑥 𝑥 3
= 5 = 2 = 1
這門課教什麼？可以用一句話
要不要講呢？
所有"未知數"的冪次項為一次。
a system of linear equations
I believe you know how to solve it.

3. In this course, m and n can be large
Review
A system of linear equations (多元一次聯立方程式)
m equations
If you can't solve many equations simultaneously then you can't solve commercial scale problems
At some point in a lot of interesting places, people get to a certain step and then say "And then it's just linear algebra from there".
questions like "How can I prepare for an interview?"  are often going to boil down to "Brush up on your linear algebra".
n variables
In this course, m and n can be large

4. Review - Terminology What actually come out of function f
Given a function f
Range (值域)
𝑣 1
𝑓 𝑣 1
𝑣 2
𝑓 𝑣 2
=𝑓 𝑣 3
Co-domain
Range
𝑣 3
Domain (定義域)
Co-domain (對應域)
What can go into function f
What may possible come out of function f

5. Review - Terminology one-to-one (一對一) Onto (映成) 𝑣 1 𝑣 1 𝑓 𝑣 1 𝑓 𝑣 1
𝑓 𝑣 1
𝑓 𝑣 1
𝑣 2
𝑣 2
𝑓 𝑣 2
𝑓 𝑣 2
=𝑓 𝑣 3
𝑣 3
𝑣 3
𝑓 𝑣 3
Co-domain = range

6. Review - Linear System Linear system have two properties
1. Persevering Multiplication
2. Persevering Addition
Linear
System
Linear
System x y kx ky
Linear
System x1 y1
x1+x2
y1+y2
Linear
System
Linear
System x2 y2

7. Question linear? Derivative: Derivative Integral from a to b linear?
function f
function f’
e.g. x2
e.g. 2x
linear?
scalar
Integral
𝑎 𝑏 𝑓 𝑡 𝑑𝑡
function f
e.g. x2
(from a to b)
e.g 𝑏 3 − 𝑎 3

8. Linear System v.s. System of Linear Equations
𝑥 1
𝑥 2
𝑥 𝑛 ……
Linear
System
𝑏 1
𝑏 2
𝑏 𝑚 ……
Domain = Rn
Co-domain = Rm ?
trivial

9. A linear system is described by a system of linear equations1 ……
𝑎 11
𝑎 21
𝑎 𝑚1 …… 1 ……
𝑎 12
𝑎 22
𝑎 𝑚2 ……
Linear
System
Linear
System 1 ……
𝑎 1𝑛
𝑎 2𝑛
𝑎 𝑚𝑛 ……
Linear
System ……
𝑥 1 …… ……
𝑥 2 …… …… …… …… ……
𝑥 𝑛
𝑥 1
𝑥 2
𝑥 𝑛 ……
Linear
System
A linear system is described by a system of linear equations