Linear Algebra (4101155 01)

Linear Algebra (4101155_01)SingLing Lee (李新林)

Tel:(05) 272-0411 X 33101

E-mail: [email protected]

Meeting: 8:45 -10:00am Mon & Wed at EA001 (After 10/03)

Office: EA407

Website: https://ecourse2.ccu.edu.tw/

Lec1-LA2021-SingLing Lee 12021/9/15

Grading

quiz : 10 %

6-8 quizzes (小考) will be given in class.

project: 3~5: 10%

mid exam #1: 20%

mid exam #2: 25% :

◼ final exam : 30%

◼Problem Discussion and Answer Questions : 5%


Office Hours

EA 407

11:00am to 12:00am on Mon and Wed.

Or by Appointment!


Textbook

”Elementary Linear Algebra : Application Version” by Howard Anton , Chris Rorres, and Anton Kaul, 12th Edition, 2019, John Wiley & Sons Inc.


Reading Textbook

understand the goal of each chapter.

read short introduction for each section.

Figure out the details of examples.

do True/False problems for each section.


Topic Outlines (I)

Matrix and Linear Equations (Chap 1, 2)

Gaussian elimination

Elementary matrix operations

Inverse Matrix

Determinants (行列式)


Topic Outlines (II)

Vector Space (Chap. 3, 4, 5)

Euclidean Vector Space

Dot product and orthogonality

Linear Independence

Basis and Dimension

Row space, column space and null space of a matrix.


Topic Outlines (III)

Inner Product Space (Chap. 6)

Orthogonality

Orthonormal Bases: QR decomposition

Best Approximation: Projection; Least Square


Topic Outlines (IV)

Diagonalization (Chap.7)

Eigenvalues and Eigenvectors

Diagonalization

Dimension Reduction

Quadratic Forms


Topic Outlines (V)

Linear Transformation (Chap.8)

General linear transformation

Kernel and range

Change of basis


Topic Outlines (VI)

Numerical Methods (Chap.9)

LU-Decomposition

The Power Method

Internet Search Engine

SVD


Other Resource

MIT Open Courseware (Yutube)

http://web.mit.edu/18.06/www/

Gilbert Strang's, Introduction to Linear Algebra, 4th edition

Videos

Homework and Exams


http://web.mit.edu/18.06/www/

http://math.mit.edu/linearalgebra/

Linear AlgebraLecture 1

• Linear Equations

• Linear System and Its Solutions

• Solving Linear Systems in Matrix Form

• Covered Range : 1.1


Selected Exercise

True-False Problems at the end of each section

(Solutions are at the end of textbook)


Linear Equations (線性方程式) (I)

A straight line in 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒 (2-dimension) :

a1x + a2y = b

a1, a2 and 𝑏 are real (實數) constants (常數). 𝑥 and 𝑦 are variables (變數).

A plane in 𝑥𝑦𝑧 − 𝑠𝑝𝑎𝑐𝑒 (3-dimension);

a1x + a2y + a3z = b

a1, a2, a3 and b are real constants. 𝑥, y and 𝑧 are variables.


Linear Equations (線性方程式)(II)

A linear equation in 𝑛 variables x1, x2, …, xn:

a1x1+ a2x2+ ∙∙∙ + anxn = b

a1, a2, …, an, b are real constants.

欲判斷是否為linear equations,看是否能化成ax+by=c的形式

(a,b,c為常數,x,y須為1次方)


Nonlinear Equations

𝑥 + 3 𝑦 = 5

𝑥𝑦 = 3

𝑥2 + 𝑦2 = 5 ;

𝑦 = sin 𝑥 ;








Linear System

Linear System：A finite set of linear equations.

A solution must satisfy each of linear equation.

Lec1-LA2021-SingLing Lee 19

4𝑥1 − 𝑥2 + 3𝑥3 = −13𝑥1 + 𝑥2 + 9𝑥3 = −4

𝑥1 = 1, 𝑥2 = 2, 𝑥3 = −1 (O)𝑥1 = 1, 𝑥2 = 8, 𝑥3 = 1 (X)⇒

2021/9/15

Linear System-No Solutin


A solution must satisfy each of linear equations.


𝑥 + 𝑦 = 42𝑥 + 2𝑦 = 6

No Solution!⇒

2021/9/15

Linear System-Infinite Solutions


A solution must satisfy each of linear equations.


ቊ4𝑥 − 2𝑦 = 116𝑥 − 8𝑦 = 4

4𝑥 − 2𝑦 = 1; (無窮多解)

𝐼𝑛𝑓𝑖𝑛𝑖𝑡𝑒 # 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠

⇒

ቊ4𝑥 − 2𝑦 = 1

0 = 0⇒

2021/9/15

Infinitely Many Solutions

Find the solution set 4x – 2y = 1 ;

(Parametric equations) ( 參數方程式)

The solution set : (x,y) =


𝑡, 2𝑡 −1

2ȁ 𝑡 ∈ 𝑅

𝑥 = 𝑡 ; 𝑦 = 2𝑡－1

2;

𝑡 =1 ( 𝑥=1, 𝑦 =3

2)

𝑡 =3 ( 𝑥=3, 𝑦 =11

2) t can be any number.

2021/9/15

Linear System General Form (一般式) :

𝑎11𝑥1 + 𝑎12𝑥2 +⋯+ 𝑎1𝑛𝑥𝑛 = 𝑏1𝑎21𝑥1 + 𝑎22𝑥2 +⋯+ 𝑎2𝑛𝑥𝑛 = 𝑏2

𝑎𝑚1𝑥1 + 𝑎𝑚2𝑥2 +⋯+ 𝑎𝑚𝑛𝑥𝑛 = 𝑏𝑚


Solution of Linear Systems

Let


(Figure 1.1.1 in pp.4)

x

yl2l1

x

y l2l1

x

yl2 l1and

(a) No solution (b) One solution (c) Infinitely manysolutions

𝑙1: 𝑎1𝑥 + 𝑏1𝑦 = 𝑐1

𝑙2: 𝑎2𝑥 + 𝑏2𝑦 = 𝑐2

2021/9/15

Solution of Linear Systems

Every linear system has exactly one of the following conditions

1) No solution

2) Exactly one solution

3) Infinitely many solution


Solve Linear Equations (I)

x – 2y = 1 -------(1)

3x + 2y = 11 -------(2)

-3．(1) + (2)

8y = 8 y = 1

x = 3

(x,y) = (3.1) is the

intersection point for two lines (equations)


⇒

⇒ ⇒

x

y

1 2 3

1

x-2y=1

3x+2y=11

x=3

y=1

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Solving Linear Equations (II)

Before : x - 2y = 1 After : x - 2y = 1

3x+ 2y =11 8y = 8


y

1 2 3

1

x-2y=1

3x+2y=11

x=3

y=1

Before elimination After elimination

y

1 2 3

1

x x

x-2y=1

8y=8

(3,1)

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Solving Linear Equations by Matrix

x – 2y = 1 -------(1)

3x + 2y = 11 -------(2)


1 −23 2

⋅𝑥𝑦 =

111

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Solving Linear Eq. by Matrix Operations

x – 2y = 1 -------(1)

3x + 2y = 11 -------(2)

Multiply -3 in 1st row then add to 2nd row


1 −23 2

⋅𝑥𝑦 =

111

1 −20 8

⋅𝑥𝑦 =

18

8𝑦 = 8, 𝑦 = 1

𝑥 − 2𝑦 = 1, 𝑥 = 32021/9/15

Example for Three Equations2x + 4y - 2z = 2 -------(1)

4x + 9y - 3z = 8 -------(2)

-2x - 3y + 7z = 10 -------(3)

(Variable x is removed in (2) and (3))

Step 1 : -2*(1) + (2)

Step 2 : (1) + (3)

2x + 4y - 2z = 2 -------(1)

y + z = 4 -------(2)’

y + 5z = 12 -------(3)’


2 4 −24 9 −3−2 −3 7

2810

2 4 −20 1 10 1 5

2412

2021/9/15

Example for Three EquationsStep 2: 2x + 4y - 2z = 2 -------(1)

y + z = 4 -------(2)’

y + 5z = 12 -------(3)’

Step 3 : Remove y in (3) : -1*(2)’ + (3)’

2x + 4y - 2z = 2 -------(1)

y + z = 4 -------(2)’

4z = 8 -------(3)’

Note : First nonzero in the row that does the elimination

2 in (1) and 1 in (2)


2 4 −20 1 10 1 5

2412

2 4 −20 1 10 0 4

248

2021/9/15

Back Substitution2x + 4y - 2z = 2 4z = 8

y + z = 4 y + z = 4

4z = 8 2x + 4y – 2z = 2

4z = 8 z = 2

y + z = 4 y + 2 = 4 y = 2

2x + 4y - 2z = 2

2x = 2 – 4y + 2z = 2 – 4*2 + 2*2 = -2 x = -1

(x , y , z) = (-1 , 2 , 2)


⇒

⇒

⇒ ⇒

⇒

2 4 −20 1 10 0 4

248

2021/9/15

Gaussian Elimination

2x + 4y - 2z = 2 -------(1)

4x + 9y - 3z = 8 -------(2)

-2x - 3y + 7z = 10 -------(3)


2 4 −24 9 −3−2 −3 7

2810

2 4 −20 1 10 0 4

248

⇒

2021/9/15

Linear System General Form

𝑎11𝑥1 + 𝑎12𝑥2 +⋯+ 𝑎1𝑛𝑥𝑛 = 𝑏1𝑎21𝑥1 + 𝑎22𝑥2 +⋯+ 𝑎2𝑛𝑥𝑛 = 𝑏2

𝑎𝑚1𝑥1 + 𝑎𝑚2𝑥2 +⋯+ 𝑎𝑚𝑛𝑥𝑛 = 𝑏𝑚


Matrix Operations

Matrix representations for linear system

Operations that will not change solutions

Status of matrices which have exact one, infinite many and none solution!

GPU computers : Fast Matrix Operations!


Conclusion

How to use computer to solve linear systems?

In practice, very common to have 1000 variables + 2000 equations.

Use Matrix + Gaussian Elimination.

Systemically : Step by Step (Algorithm)

Fast : (Computing Efficiency)

Accuracy : (Numerical Analysis)


Applications

Machine Learning

Graphis, Image Processing, Vision

Deep Learning

Artificial Intelligence

Data Science

….


Documents

Linear Algebra (4101155 01)