Linear Algebra 2021 Fall



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Exercises

DueDate Topic YouTube PDF  PPT 
10/ 1 Exercise 1.6 (Oct. 1) PDF PPT
10/ 1 Exercise 1.7 (Oct. 1) PDF PPT
10/ 8 Exercise 1.7.2 (Oct. 8) PDF PPT
10/ 8 Exercise 1.4 (Oct. 8) PDF PPT
10/ 8 Exercise 2.1 (Oct. 8) PDF PPT
10/15 Exercise 2.1 (Oct. 15) PDF PPT
10/15 Exercise 2.3 (Oct. 15) PDF PPT
10/22 Exercise 2.4 (Oct. 22) PDF PPT
10/22 Exercise 4.1 (Oct. 22) PDF PPT
10/29 Exercise 4.1 (Oct. 29) PDF PPT
10/29 Exercise 4.2 (Oct. 29) PDF PPT
10/29 Exercise 4.3 (Oct. 29) PDF PPT
11/ 5 Exercise 4.3 (Nov. 5) PDF PPT
11/ 5 Chapter 3 (Nov. 5) PDF PPT
11/ 5 Review (Nov. 5) PDF PPT
11/19 Exercise 4.4 (Nov. 19) Video PDF PPT
11/19 Exercise 4.5 (Nov. 19) Video PDF PPT😢
11/26 Exercise 5.1 (Nov. 26) Video (former half) PDF PPT
11/26 Exercise 5.2 (Nov. 26) Video (latter half) PDF PPT

😢 : Sorry for missing some handwriting.
🚧 : Under construction.



Course Materials



review 🔍overview basic concept optional
def. definition ex. example pf. proof thm. theorem
review
🔍overview
basic concept
optional
def. definition
ex. example
pf. proof
thm. theorem
Logistics
----- Course Policy ----- YouTube PDF PPT
Chapter 1
DueDate Topic YouTube Textbook PDF PPT
1 linear
9/22 def. Linear System
9/22 ex. Are they Linear System?
9/22 ex. Derivative and Integral are Linear Systems
2 course introduction
yourself Linear Algebra v.s. Compulsory Courses (optional)
yourself Course Overview (optional)
3 vector
yourself Vector 1.1
yourself Properties of Vector 1.1
4 system of linear equations
9/24 System of Linear Equations 1.3
9/24 System of Linear Equations = Linear System 1.3
5 matrix
9/24 Matrix 1.1
9/24 Properties of Matrix 1.1
9/24 def. Diagonal, Identity, Zero Matrix 1.2
9/24 def. Transpose 1.1
4 system of linear equations
9/24 Matrix-Vector Product 1.2
9/24 Matrix-Vector Product = System of Linear Equations 1.2
9/24 ex. Matrix-Vector Product (Example) 1.2
9/24 Properties of Matrix-Vector Product 1.2
9/24 Standard Vector 1.2
6 solution
9/24 Solution of System of Linear Equations (high school) 1.3
9/24 🔍 Solution of System of Linear Equations (this course)
9/24 def. Linear Combination 1.2
9/24 Linear Combination v.s. Solution 1.2
9/24 ex. Linear Combination v.s. Solution (Example 1) 1.2
9/24 ex. Linear Combination v.s. Solution (Example 2) 1.2
9/24 ex. Linear Combination v.s. Solution (Example 3) 1.2
9/29 def. Span 1.6
9/29 ex. Span (Example) 1.6
9/29 Span v.s. Solution 1.6
9/29 thm. Span (Theorem of Useless Vector) 1.6
9/29 pf. Span (Theorem of Useless Vector) 1.6
10/ 1 def. Dependent / Independent 1.7
10/ 1 ex. Dependent / Independent (Example) 1.7
10/ 1 Dependent / Independent (Intuitive Explaination) 1.7
10/ 1 Dependent / Independent v.s. Solution 1.7
10/ 1 ex. Dependent / Independent v.s. Solution (Example) 1.7
10/ 1 def. Dependent / Independent (Another Definition) 1.7
10/ 1 pf. Dependent / Independent v.s. Solution (Proof) 1.7
10/ 1 def. Rank / Nullity
10/ 1 ex. Rank / Nullity (Example 1)
10/ 1 ex. Rank / Nullity (Example 2)
10/ 1 ex. Rank / Nullity (Example 3)
10/ 1 Rank / Nullity v.s. Solution
10/ 1 Story of Gaussian Elimination (optional) 1.4
10/ 1 Strategy of Finding Solutions 1.4
10/ 1 Elementary Row Operation 1.4
10/ 1 def. REF 1.4
10/ 1 def. RREF 1.4
10/ 1 def. Pivot Columns 1.4
10/ 1 thm. RREF is unique 1.4
10/ 1 RREF v.s. unique solution 1.4
10/ 1 RREF v.s. infinite solutions 1.4
10/ 1 RREF v.s. no solution 1.4
10/ 1 ~~~~~~ HW1 Released! ~~~~~~ Go to...
10/ 6 ex. Find RREF (Example 1) 1.4
10/ 6 ex. Find RREF (Example 1) - Find solution 1.4
10/ 6 ex. Find RREF (Example 2) 1.4
10/ 6 ex. Find RREF (Example 3) 1.4
7 RREF
10/ 6 thm. Column Correspondence Theorem
10/ 6 Column Correspondence Theorem - Reason 1
10/ 6 thm. Ax = 0 and Rx = 0 are equivalent
10/ 6 Column Correspondence Theorem - Reason 2
10/ 6 No Row Correspondence Theorem
10/ 8 How to Check Independence 1.7
10/ 8 Independence v.s. Column Correspondence Theorem 1.7
10/ 8 Independence v.s. Matrix Size 1.7
10/ 8 def. Rank = no. of Pivot Columns = no. of non-zero rows in RREF 1.7
10/ 8 Independence v.s. Matrix Size (again) 1.7
10/ 8 def. Rank v.s. Basic / Free Variables 1.7
10/ 8 🔍 Definitions of Rank and Nullity 1.7
10/ 8 All properties about always consistent 1.7
10/ 8 thm. More than m vectors in Rm must be dependent 1.7
10/ 8 Three is a powerful number :) (optional) 1.7
Chapter 2
DueDate Topic YouTube Textbook PDF PPT
1 matrix multiplication
10/ 8 Matrix Multiplication: inner product 2.1
10/ 8 Matrix Multiplication: Combination of Columns 2.1
10/ 8 Matrix Multiplication: Combination of Rows 2.1
10/ 8 Matrix Multiplication: Summation of Matrices 2.1
10/ 8 Block Multiplication 2.1
10/ 8 ex. Block Multiplication - Example 2.1
10/ 8 Matrix Multiplication means multiple inputs 2.1
10/ 8 Matrix Multiplication represents Composition 2.1
10/ 8 ex. Matrix Multiplication represents Composition - Example 2.1
10/13 Matrix Multiplication - Properties 2.1
10/13 Matrix Multiplication - Transpose 2.1
10/13 Matrix Multiplication - Pratical Computation Issue (optional) 2.1
2 matrix inverse
10/13 def. Inverse of Matrix 2.4
10/13 Inverse of Matrix - Properties 2.4
10/13 Inverse of Matrix - Matrix Transpose 2.4
10/13 Inverse of Matrix - Matrix Multiplication 2.4
10/15 Inverse of Matrix - Solving System of Linear Equations (optional) 2.4
10/15 Inverse of Matrix - Input-output Model 1 (optional) 2.4
10/15 Inverse of Matrix - Input-output Model 2 (optional) 2.4
10/15 thm. Invertible Matrix Theorem 2.4
10/15 Review: one-to-one and onto 2.8
10/15 One-to-one in Linear Algebra 2.8
10/15 Onto in Linear Algebra 2.8
10/15 Invertible = One-to-one and Onto 2.8
10/15 pf. Invertible Matrix Theorem - Proof (part 1) 2.4
10/15 pf. Invertible Matrix Theorem - Proof (part 2) 2.4
10/15 def. Elementary Matrix 2.3
10/15 Inverse of Elementary Matrix 2.3
10/15 pf. Invertible Matrix Theorem - Proof (part 3) 2.4
10/15 Find A-1 (Special Case: 2x2 matrices) (optional) 2.4
10/15 Find A-1 2.4
10/15 Find A-1C 2.4
10/15 ~~~~~~ HW2 Released! ~~~~~~ Go to...
Chapter 4
DueDate Topic YouTube Textbook PDF PPT
  subspace
10/20 def. Subspace 4.1
10/20 ex. Subspace - Example 4.1
10/20 Subspace v.s. Span 4.1
10/20 def. Column Space and Row Space 4.3
10/20 def. Null Space 4.3
10/22 def. Basis 4.2
10/22 ex. Basis - Example 4.2
10/22 thm. More Theorems of Span 4.2
10/22 thm. Three Theorems of Basis 4.2
10/22 def. Dimension 4.2
10/22 More than m vectors in Rm must be dependent (again and again) 4.2
10/22 pf. Proof of Basis Theorem 1 - Reduction Theorem 4.2
10/22 pf. Proof of Basis Theorem 2 - Extension Theorem 4.2
10/22 pf. Proof of Basis Theorem 3 - Dimension 4.2
10/22 thm. Dimension v.s. "Size" of Subspace 4.3
10/22 🔍 Three Theorems of Basis (review) 4.2
10/22 Is it a basis? - Based on Definition 4.2
10/22 Is it a basis? - Easier Way 4.2
10/22 ex. Is it a basis? - Example 4.2
10/22 Basis and Dimension of Column Space (More definitions of Rank!) 4.3
10/22 Basis and Dimension of Row Space (More definitions of Rank!) 4.3
10/22 thm. Rank A = Rank AT !!! 4.3
10/22 Basis and Dimension of Null Space 4.3
10/22 thm. Dimension Theorem 4.3
10/27 def. Coordinate System 4.4
10/27 ex. Coordinate System - Example 4.4
10/27 莊子齊物論 (optional) 4.4
10/27 def. Cartesian Coordinate System 4.4
10/27 蓋亞思維 (optional) 4.4
10/27 A coordinate system is a basis 4.4
10/27 Other system to Cartesian 4.4
10/27 Cartesian to Other system 4.4
10/27 Change Coordinate 4.4
10/29 Equation of ellipse (optional) 4.4
10/29 Equation of hyperbola (optional) 4.4
10/29 全面啟動 (optional) 4.5
10/29 ex. Describing a function in another coordinate system 4.5
10/29 Function in Different Coordinate Systems 4.5
10/29 ex. Function in Different Coordinate Systems - Example 4.5
10/29 ex. Function in Different Coordinate Systems - Example 4.5
Chapter 3
DueDate Topic YouTube Textbook PDF PPT
  determinant
10/29 Determinant (high school) (optional) 3.1
10/29 def. Determinant - Cofactor Expansion 3.1
10/29 ex. Determinant of 2x2 and 3x3 matrices 3.1
10/29 ex. Determinant of 2x2 and 3x3 matrices 3.1
10/29 Determinant of a special gigantic matrix (optional) 3.1
11/ 3 def. Three Basic Properties of Determinant
11/ 3 Basic Property 1
11/ 3 Basic Property 2
11/ 3 Basic Property 3
11/ 3 From Basic Properties to Cofactor Expansion (2x2 matrix) (optional)
11/ 3 From Basic Properties to Cofactor Expansion (3x3 matrix) (optional)
11/ 3 From Basic Properties to Cofactor Expansion (nxn matrix) (optional)
11/ 5 Formula of A-1 (optional)
11/ 5 Formula of A-1 - Example (optional)
11/ 5 Formula of A-1 - Proof (optional)
11/ 5 Cramer’s Rule (optional) 3.2
11/ 5 Three Basic Properties of Determinant (review) (optional) 3.2
11/ 5 thm. A is invertible = det (A) is not zero 3.2
11/ 5 ex. example 3.2
11/ 5 thm. Properties of Determinant 3.2
11/ 5 pf. det(AB) = det(A)det(B) 3.2
11/ 5 pf. det(A) = det (AT) 3.2
11/ 5 ~~~~~~ HW3 Released! ~~~~~~ Go to...
Chapter 5
DueDate Topic YouTube Textbook PDF PPT
  eigenvalues and eigenvectors
11/24 How to find a "good" coordinate system? (optional) 5.1
11/24 def. Eigenvalues and Eigenvectors 5.1
11/24 ex. Example 5.1
11/24 Do the eigenvectors correspond to an eigenvalue from a subspace? 5.1
11/24 def. Eigenspace 5.1
11/24 Check whether a scalar is an eigenvalue 5.1
11/24 ex. Example 5.1
11/24 Looking for Eigenvalues 5.1
11/24 ex. Looking for Eigenvalues - Example 1 5.1
11/26 ex. Looking for Eigenvalues - Example 2 5.1
11/26 ex. Looking for Eigenvalues - Example 3 5.1
11/26 def. Characteristic Polynomial 5.2
11/26 Matrix A and RREF of A have different eigenvalues 5.2
11/26 thm. Similar matrices have the same eigenvalues 5.2
11/26 thm. More Properties of Characteristic Polynomial 5.2
11/26 PageRank: How does Google rank search results? (optional)
11/26 PageRank: Introduction (optional)
11/26 PageRank: Basic Idea (optional)
11/26 PageRank: Formulation (optional)
11/26 PageRank: Relation to Eigenvectors / Eigenvalues (optional)
11/26 PageRank: Always having eigenvalue = 1 (optional)
11/26 PageRank: When does dimension of eigenspace = 1 (optional)
11/26 PageRank: How to make dimension of eigenspace = 1 (optional)
11/26 PageRank: Power Method (optional)
12/ 1 def. Diagonalizable 5.3
12/ 1 Not all matrices are diagonalizable 5.3
12/ 1 How to diagonalize a matrix 5.3
12/ 1 thm. Eigenvectors corresponding to distinct Eigenvalues is independent 5.3
12/ 1 Find independent eigenvectors 5.3
12/ 1 ex. Example 5.3
12/ 1 Test for Diagonalizable Matrix 5.3
12/ 1 Application of Diagonalization 1: 這就是人生! (optional) 5.3
12/ 1 Application of Diagonalization 1: 你花了多少時間在念線性代數? (optional) 5.3
12/ 1 ex. Diagonalization of Linear Operator 5.3
12/ 1 Application of Diagonalization 2: Find a good Coordinate System 5.3
Chapter 7
DueDate Topic YouTube Textbook PDF PPT
  orthogonality
12/ 8 def. Norm and Distance 7.1
12/ 8 def. Dot Product and Orthogonal 7.1
12/ 8 thm. Pythagorean Theorem 7.1
12/ 8 thm. Dot Product v.s. Geometry 7.1
12/ 8 thm. Triangle Inequality 7.1
12/10 def. Orthogonal Set 7.2
12/10 Orthogonal Set v.s. Independent Set 7.2
12/10 def. Orthonormal Set 7.2
12/10 def. Orthogonal / Orthonormal Basis 7.2
12/10 thm. Orthogonal Decomposition Theory 7.2
12/10 ex. Example 7.2
12/10 thm. Gram-Schmidt Process 7.2
12/10 ex. Example 7.2
12/10 pf. Proof of Gram-Schmidt Process (1): Obtaining Orthogonal Set 7.2
12/10 pf. Proof of Gram-Schmidt Process (2): Obtaining Basis 7.2
12/10 def. Orthogonal Complement 7.3
12/10 ex. Example 7.3
12/10 thm. B be a basis of W, then B = W 7.3
12/10 ex. How to find W 7.3
12/10 thm. Orthogonal Complement v.s. Null Space 7.3
12/10 thm. u = w + z → w ∈ W, z ∈ W 7.3
12/15 Orthogonal Projection 7.4
12/15 thm. Closest Vector Property 7.4
12/15 Orthogonal Projection Matrix 7.4

Homework

# Date Topic TA Slides Video
10/ 1 Colab Tutorial 陳建成 Slides Video
HW110/ 1 Cycle Detection 林泓均 Slides Video
HW210/15 Hill Cipher 劉聿珉 Slides Video (original)
HW311/ 5 Cosine Transform and Its Application 林冠廷 Slides Video
HW411/26 PageRank 翁茂齊 Slides Video (original)
HW512/17  (Coming soon...)  (Guess who?) (Coming soon...) (Coming soon...)
HW6 1/14  (Coming soon...)  (Guess who?) (Coming soon...) (Coming soon...)
Note: 🚧  implies the material is not available now.