Linear Algebra (4101155 01)
Transcript of 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/
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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%
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Office Hours
EA 407
11:00am to 12:00am on Mon and Wed.
Or by Appointment!
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Textbook
”Elementary Linear Algebra : Application Version” by Howard Anton , Chris Rorres, and Anton Kaul, 12th Edition, 2019, John Wiley & Sons Inc.
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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.
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Topic Outlines (I)
Matrix and Linear Equations (Chap 1, 2)
Gaussian elimination
Elementary matrix operations
Inverse Matrix
Determinants (行列式)
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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.
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Topic Outlines (III)
Inner Product Space (Chap. 6)
Orthogonality
Orthonormal Bases: QR decomposition
Best Approximation: Projection; Least Square
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Topic Outlines (IV)
Diagonalization (Chap.7)
Eigenvalues and Eigenvectors
Diagonalization
Dimension Reduction
Quadratic Forms
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Topic Outlines (V)
Linear Transformation (Chap.8)
General linear transformation
Kernel and range
Change of basis
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Topic Outlines (VI)
Numerical Methods (Chap.9)
LU-Decomposition
The Power Method
Internet Search Engine
SVD
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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
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Linear AlgebraLecture 1
• Linear Equations
• Linear System and Its Solutions
• Solving Linear Systems in Matrix Form
• Covered Range : 1.1
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Selected Exercise
True-False Problems at the end of each section
(Solutions are at the end of textbook)
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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.
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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)
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Nonlinear Equations
𝑥 + 3 𝑦 = 5
𝑥𝑦 = 3
𝑥2 + 𝑦2 = 5 ;
𝑦 = sin 𝑥 ;
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Linear AlgebraLecture 1
• Linear Equations
• Linear System and Its Solutions
• Solving Linear Systems in Matrix Form
• Covered Range : 1.1
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Linear System
Linear System:A finite set of linear equations.
A solution must satisfy each of linear equation.
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4𝑥1 − 𝑥2 + 3𝑥3 = −13𝑥1 + 𝑥2 + 9𝑥3 = −4
𝑥1 = 1, 𝑥2 = 2, 𝑥3 = −1 (O)𝑥1 = 1, 𝑥2 = 8, 𝑥3 = 1 (X)⇒
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Linear System-No Solutin
Linear System:A finite set of linear equations.
A solution must satisfy each of linear equations.
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𝑥 + 𝑦 = 42𝑥 + 2𝑦 = 6
No Solution!⇒
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Linear System-Infinite Solutions
Linear System:A finite set of linear equations.
A solution must satisfy each of linear equations.
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ቊ4𝑥 − 2𝑦 = 116𝑥 − 8𝑦 = 4
4𝑥 − 2𝑦 = 1; (無窮多解)
𝐼𝑛𝑓𝑖𝑛𝑖𝑡𝑒 # 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠
⇒
ቊ4𝑥 − 2𝑦 = 1
0 = 0⇒
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Infinitely Many Solutions
Find the solution set 4x – 2y = 1 ;
(Parametric equations) ( 參數方程式)
The solution set : (x,y) =
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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.
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Linear System General Form (一般式) :
𝑎11𝑥1 + 𝑎12𝑥2 +⋯+ 𝑎1𝑛𝑥𝑛 = 𝑏1𝑎21𝑥1 + 𝑎22𝑥2 +⋯+ 𝑎2𝑛𝑥𝑛 = 𝑏2
𝑎𝑚1𝑥1 + 𝑎𝑚2𝑥2 +⋯+ 𝑎𝑚𝑛𝑥𝑛 = 𝑏𝑚
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Solution of Linear Systems
Let
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(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
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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
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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)
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⇒
⇒ ⇒
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
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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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Linear AlgebraLecture 1
• Linear Equations
• Linear System and Its Solutions
• Solving Linear Systems in Matrix Form
• Covered Range : 1.1
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Solving Linear Equations by Matrix
x – 2y = 1 -------(1)
3x + 2y = 11 -------(2)
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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
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1 −23 2
⋅𝑥𝑦 =
111
1 −20 8
⋅𝑥𝑦 =
18
8𝑦 = 8, 𝑦 = 1
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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)’
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2 4 −24 9 −3−2 −3 7
2810
2 4 −20 1 10 1 5
2412
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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)
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2 4 −20 1 10 1 5
2412
2 4 −20 1 10 0 4
248
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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)
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⇒
⇒
⇒ ⇒
⇒
2 4 −20 1 10 0 4
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Gaussian Elimination
2x + 4y - 2z = 2 -------(1)
4x + 9y - 3z = 8 -------(2)
-2x - 3y + 7z = 10 -------(3)
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2 4 −24 9 −3−2 −3 7
2810
2 4 −20 1 10 0 4
248
⇒
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Linear System General Form
𝑎11𝑥1 + 𝑎12𝑥2 +⋯+ 𝑎1𝑛𝑥𝑛 = 𝑏1𝑎21𝑥1 + 𝑎22𝑥2 +⋯+ 𝑎2𝑛𝑥𝑛 = 𝑏2
𝑎𝑚1𝑥1 + 𝑎𝑚2𝑥2 +⋯+ 𝑎𝑚𝑛𝑥𝑛 = 𝑏𝑚
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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!
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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)
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Applications
Machine Learning
Graphis, Image Processing, Vision
Deep Learning
Artificial Intelligence
Data Science
….
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