Linear Algebra 2022 Fall



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2022 ESSENCE

Topic VIDEO SLIDE
System of Linear Equations
AlphaTensor: 用增強式學習找出更有效率的矩陣相乘演算法
期中考總複習 (2022)
review CH4: Coordinate System & Eigenvalues/Eigenvectors
Review Diagonalization
Review Orthogonal
Review Orthogonal Projection
用 Chat GPT 解線性代數考古題


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
Chapter 1
DueDate Topic YouTube Textbook PDF PPT
1 linear
9/9 def. Linear System
9/9 ex. Are they Linear System?
9/9 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
9/16 System of Linear Equations 1.3
9/16 System of Linear Equations = Linear System 1.3
5 matrix
9/16 Matrix 1.1
9/16 Properties of Matrix 1.1
9/16 def. Diagonal, Identity, Zero Matrix 1.2
9/16 def. Transpose 1.1
4 system of linear equations
9/16 Matrix-Vector Product 1.2
9/16 Matrix-Vector Product = System of Linear Equations 1.2
9/16 ex. Matrix-Vector Product (Example) 1.2
9/16 Properties of Matrix-Vector Product 1.2
9/16 Standard Vector 1.2
6 solution
9/16 Solution of System of Linear Equations (high school) 1.3
9/16 🔍 Solution of System of Linear Equations (this course)
9/16 def. Linear Combination 1.2
9/16 Linear Combination v.s. Solution 1.2
9/16 ex. Linear Combination v.s. Solution (Example 1) 1.2
9/16 ex. Linear Combination v.s. Solution (Example 2) 1.2
9/16 ex. Linear Combination v.s. Solution (Example 3) 1.2
9/16 def. Span 1.6
9/16 ex. Span (Example) 1.6
9/16 Span v.s. Solution 1.6
9/16 thm. Span (Theorem of Useless Vector) 1.6
9/16 pf. Span (Theorem of Useless Vector) 1.6
9/23 Exercise 1.6
9/23 def. Dependent / Independent 1.7
9/23 ex. Dependent / Independent (Example) 1.7
9/23 Dependent / Independent (Intuitive Explaination) 1.7
9/23 Dependent / Independent v.s. Solution 1.7
9/23 ex. Dependent / Independent v.s. Solution (Example) 1.7
9/23 def. Dependent / Independent (Another Definition) 1.7
9/23 pf. Dependent / Independent v.s. Solution (Proof) 1.7
9/23 Exercise 1.7
9/23 Exercise 1.7.2
9/23 def. Rank / Nullity
9/23 ex. Rank / Nullity (Example 1)
9/23 ex. Rank / Nullity (Example 2)
9/23 ex. Rank / Nullity (Example 3)
9/23 Rank / Nullity v.s. Solution
9/23 Story of Gaussian Elimination (optional) 1.4
9/23 Strategy of Finding Solutions 1.4
9/23 Elementary Row Operation 1.4
9/23 def. REF 1.4
9/23 def. RREF 1.4
9/23 def. Pivot Columns 1.4
9/23 thm. RREF is unique 1.4
9/23 RREF v.s. unique solution 1.4
9/23 RREF v.s. infinite solutions 1.4
9/23 RREF v.s. no solution 1.4
9/23 ~~~~~~ HW1 Released! ~~~~~~ Go to...
9/30 ex. Find RREF (Example 1) 1.4
9/30 ex. Find RREF (Example 1) - Find solution 1.4
9/30 ex. Find RREF (Example 2) 1.4
9/30 ex. Find RREF (Example 3) 1.4
9/30 Exercise 1.4
7 RREF
9/30 thm. Column Correspondence Theorem
9/30 Column Correspondence Theorem - Reason 1
9/30 thm. Ax = 0 and Rx = 0 are equivalent
9/30 Column Correspondence Theorem - Reason 2
9/30 No Row Correspondence Theorem
9/30 How to Check Independence 1.7
9/30 Independence v.s. Column Correspondence Theorem 1.7
9/30 Independence v.s. Matrix Size 1.7
9/30 def. Rank = no. of Pivot Columns = no. of non-zero rows in RREF 1.7
9/30 Independence v.s. Matrix Size (again) 1.7
9/30 def. Rank v.s. Basic / Free Variables 1.7
9/30 🔍 Definitions of Rank and Nullity 1.7
9/30 All properties about always consistent 1.7
9/30 thm. More than m vectors in Rm must be dependent 1.7
9/30 Three is a powerful number :) (optional) 1.7
Chapter 2
DueDate Topic YouTube Textbook PDF PPT
1 matrix multiplication
9/30 Matrix Multiplication: inner product 2.1
9/30 Matrix Multiplication: Combination of Columns 2.1
9/30 Matrix Multiplication: Combination of Rows 2.1
9/30 Matrix Multiplication: Summation of Matrices 2.1
9/30 Block Multiplication 2.1
9/30 ex. Block Multiplication - Example 2.1
9/30 Matrix Multiplication means multiple inputs 2.1
9/30 Matrix Multiplication represents Composition 2.1
9/30 ex. Matrix Multiplication represents Composition - Example 2.1
10/ 7 Matrix Multiplication - Properties 2.1
10/ 7 Matrix Multiplication - Transpose 2.1
10/ 7 Matrix Multiplication - Pratical Computation Issue (optional) 2.1
10/ 7 Exercise 2.1
2 matrix inverse
10/ 7 def. Inverse of Matrix 2.4
10/ 7 Inverse of Matrix - Properties 2.4
10/ 7 Inverse of Matrix - Matrix Transpose 2.4
10/ 7 Inverse of Matrix - Matrix Multiplication 2.4
10/ 7 Inverse of Matrix - Solving System of Linear Equations (optional) 2.4
10/ 7 Inverse of Matrix - Input-output Model 1 (optional) 2.4
10/ 7 Inverse of Matrix - Input-output Model 2 (optional) 2.4
10/ 7 thm. Invertible Matrix Theorem 2.4
10/ 7 Review: one-to-one and onto 2.8
10/ 7 One-to-one in Linear Algebra 2.8
10/ 7 Onto in Linear Algebra 2.8
10/ 7 Invertible = One-to-one and Onto 2.8
10/ 7 pf. Invertible Matrix Theorem - Proof (part 1) 2.4
10/ 7 pf. Invertible Matrix Theorem - Proof (part 2) 2.4
10/ 7 def. Elementary Matrix 2.3
10/ 7 Exercise 2.3
10/ 7 Inverse of Elementary Matrix 2.3
10/ 7 pf. Invertible Matrix Theorem - Proof (part 3) 2.4
10/ 7 Find A-1 (Special Case: 2x2 matrices) (optional) 2.4
10/ 7 Find A-1 2.4
10/ 7 Find A-1C 2.4
10/ 7 Exercise 2.4
10/ 7 ~~~~~~ HW1 Due! ~~~~~~ Go to...
10/ 7 ~~~~~~ HW2 Released! ~~~~~~ Go to...
Chapter 4-(1)
DueDate Topic YouTube Textbook PDF PPT
  subspace
10/14 def. Subspace 4.1
10/14 ex. Subspace - Example 4.1
10/14 Subspace v.s. Span 4.1
10/14 Exercise 4.1-1 PDF PPT
10/14 Exercise 4.1-2 PDF PPT
10/14 def. Column Space and Row Space 4.3
10/14 def. Null Space 4.3
10/14 def. Basis 4.2
10/14 ex. Basis - Example 4.2
10/14 thm. More Theorems of Span 4.2
10/14 thm. Three Theorems of Basis 4.2
10/14 def. Dimension 4.2
10/14 More than m vectors in Rm must be dependent (again and again) 4.2
10/14 pf. Proof of Basis Theorem 1 - Reduction Theorem 4.2
10/14 pf. Proof of Basis Theorem 2 - Extension Theorem 4.2
10/14 pf. Proof of Basis Theorem 3 - Dimension 4.2
10/14 thm. Dimension v.s. "Size" of Subspace 4.3
10/14 🔍 Three Theorems of Basis (review) 4.2
10/14 Is it a basis? - Based on Definition 4.2
10/14 Is it a basis? - Easier Way 4.2
10/14 ex. Is it a basis? - Example 4.2
10/14 Exercise 4.2 PDF PPT
10/14 Basis and Dimension of Column Space (More definitions of Rank!) 4.3
10/14 Basis and Dimension of Row Space (More definitions of Rank!) 4.3
10/14 thm. Rank A = Rank AT !!! 4.3
10/14 Basis and Dimension of Null Space 4.3
10/14 thm. Dimension Theorem 4.3
10/21 Exercise 4.3-1 PDF PPT
10/21 Exercise 4.3-2 PDF PPT
Chapter 3
DueDate Topic YouTube Textbook PDF PPT
  determinant
10/21 Determinant (high school) (optional) 3.1
10/21 def. Determinant - Cofactor Expansion 3.1
10/21 ex. Determinant of 2x2 and 3x3 matrices 3.1
10/21 ex. Determinant of 2x2 and 3x3 matrices 3.1
10/21 Determinant of a special gigantic matrix (optional) 3.1
10/21 def. Three Basic Properties of Determinant
10/21 Basic Property 1
10/21 Basic Property 2
10/21 Basic Property 3
10/21 From Basic Properties to Cofactor Expansion (2x2 matrix) (optional)
10/21 From Basic Properties to Cofactor Expansion (3x3 matrix) (optional)
10/21 From Basic Properties to Cofactor Expansion (nxn matrix) (optional)
10/21 Formula of A-1 (optional)
10/21 Formula of A-1 - Example (optional)
10/21 Formula of A-1 - Proof (optional)
10/21 Cramer’s Rule (optional) 3.2
10/21 Three Basic Properties of Determinant (review) (optional) 3.2
10/21 thm. A is invertible = det (A) is not zero 3.2
10/21 ex. example 3.2
10/21 thm. Properties of Determinant 3.2
10/21 pf. det(AB) = det(A)det(B) 3.2
10/21 pf. det(A) = det (AT) 3.2
10/21 Chapter 3 PDF PPT
10/21 Review PDF PPT
Chapter 4-(2)
  subspace
11/ 4 def. Coordinate System 4.4
11/ 4 ex. Coordinate System - Example 4.4
11/ 4 莊子齊物論 (optional) 4.4
11/ 4 def. Cartesian Coordinate System 4.4
11/ 4 蓋亞思維 (optional) 4.4
11/ 4 A coordinate system is a basis 4.4
11/ 4 Other system to Cartesian 4.4
11/ 4 Cartesian to Other system 4.4
11/ 4 Change Coordinate 4.4
11/ 4 Equation of ellipse (optional) 4.4
11/ 4 Equation of hyperbola (optional) 4.4
11/ 4 Exercise 4.4 PDF PPT
11/ 4 全面啟動 (optional) 4.5
11/ 4 ex. Describing a function in another coordinate system 4.5
11/ 4 Function in Different Coordinate Systems 4.5
11/ 4 ex. Function in Different Coordinate Systems - Example 4.5
11/ 4 ex. Function in Different Coordinate Systems - Example 4.5
11/ 4 Exercise 4.5 PDF PPT
11/ 4 ~~~~~~ HW2 Due! ~~~~~~ Go to...
11/ 4 ~~~~~~ HW3 Released! ~~~~~~ Go to...
Chapter 5
DueDate Topic YouTube Textbook PDF PPT
  eigenvalues and eigenvectors
11/11 How to find a "good" coordinate system? (optional) 5.1
11/11 def. Eigenvalues and Eigenvectors 5.1
11/11 ex. Example 5.1
11/11 Do the eigenvectors correspond to an eigenvalue from a subspace? 5.1
11/11 def. Eigenspace 5.1
11/11 Check whether a scalar is an eigenvalue 5.1
11/11 ex. Example 5.1
11/11 Looking for Eigenvalues 5.1
11/11 ex. Looking for Eigenvalues - Example 1 5.1
11/18 ex. Looking for Eigenvalues - Example 2 5.1
11/18 ex. Looking for Eigenvalues - Example 3 5.1
11/18 Exercise 5.1 PDF PPT
11/18 def. Characteristic Polynomial 5.2
11/18 Matrix A and RREF of A have different eigenvalues 5.2
11/18 thm. Similar matrices have the same eigenvalues 5.2
11/18 thm. More Properties of Characteristic Polynomial 5.2
11/18 Exercise 5.2-1 PDF PPT
11/18 Exercise 5.2-2 PDF
11/18 PageRank: How does Google rank search results? (optional)
11/18 PageRank: Introduction (optional)
11/18 PageRank: Basic Idea (optional)
11/18 PageRank: Formulation (optional)
11/18 PageRank: Relation to Eigenvectors / Eigenvalues (optional)
11/18 PageRank: Always having eigenvalue = 1 (optional)
11/18 PageRank: When does dimension of eigenspace = 1 (optional)
11/18 PageRank: How to make dimension of eigenspace = 1 (optional)
11/18 PageRank: Power Method (optional)
11/18 ~~~~~~ HW3 Due! ~~~~~~ Go to...
11/18 ~~~~~~ HW4 Released! ~~~~~~ Go to...
11/18 def. Diagonalizable 5.3
11/18 Not all matrices are diagonalizable 5.3
11/18 How to diagonalize a matrix 5.3
11/18 thm. Eigenvectors corresponding to distinct Eigenvalues is independent 5.3
11/18 Find independent eigenvectors 5.3
11/18 ex. Example 5.3
11/18 Test for Diagonalizable Matrix 5.3
11/18 Application of Diagonalization 1: 這就是人生! (optional) 5.3
11/18 Application of Diagonalization 1: 你花了多少時間在念線性代數? (optional) 5.3
11/18 ex. Diagonalization of Linear Operator 5.3
11/18 Application of Diagonalization 2: Find a good Coordinate System 5.3
11/25 Exercise 5.3 PDF
11/25 Chapter 5 PDF
Chapter 7
DueDate Topic YouTube Textbook PDF PPT
  orthogonality
11/25 def. Norm and Distance 7.1
11/25 def. Dot Product and Orthogonal 7.1
11/25 thm. Pythagorean Theorem 7.1
11/25 thm. Dot Product v.s. Geometry 7.1
11/25 thm. Triangle Inequality 7.1
11/25 def. Orthogonal Set 7.2
11/25 Orthogonal Set v.s. Independent Set 7.2
11/25 def. Orthonormal Set 7.2
11/25 def. Orthogonal / Orthonormal Basis 7.2
11/25 thm. Orthogonal Decomposition Theory 7.2
11/25 ex. Example 7.2
11/25 thm. Gram-Schmidt Process 7.2
11/25 ex. Example 7.2
11/25 pf. Proof of Gram-Schmidt Process (1): Obtaining Orthogonal Set 7.2
11/25 pf. Proof of Gram-Schmidt Process (2): Obtaining Basis 7.2
11/25 def. Orthogonal Complement 7.3
11/25 ex. Example 7.3
11/25 thm. B be a basis of W, then B = W 7.3
11/25 ex. How to find W 7.3
11/25 thm. Orthogonal Complement v.s. Null Space 7.3
11/25 thm. u = w + z → w ∈ W, z ∈ W 7.3
12/ 2 def. Orthogonal Projection 7.4
12/ 2 thm. Closest Vector Property 7.4
12/ 2 def. Orthogonal Projection Matrix 7.3
12/ 2 Orthogonal Projection on a line 7.3
12/ 2 thm. Orthogonal Projection Matrix 7.3
12/ 2 pf. Orthogonal Projection Matrix - Proof (part I) 7.3
12/ 2 pf. Orthogonal Projection Matrix - Proof (part II) 7.3
12/ 2 Orthogonal Decomposition Theory v.s. Orthogonal Projection Matrix 7.3
12/ 2 Exercise 7.3 PDF PPT
12/ 2 Applications of Orthogonal Projection 7.4
12/ 2 Least Square Approximation - Problem Statement 7.4
12/ 2 Least Square Approximation - Solving by Orthogonal Projection 7.4
12/ 2 ex. Least Square Approximation - Example 1 7.4
12/ 2 ex. Least Square Approximation - Example 2 7.4
12/ 2 ex. Least Square Approximation - Example 3 7.4
12/ 9 Exercise 7.4 PDF PPT
12/ 2 def. Orthogonal Matrix 7.5
12/ 2 def. Norm-preserving 7.5
12/ 2 Orthogonal Matrix = Norm-preserving 7.5
12/ 2 thm. Properties of Orthogonal Matrix 7.5
12/ 2 pf. Properties of Orthogonal Matrix - Proof 7.5
12/ 2 thm. det Q, PQ, Q⁻¹, Qᵀ 7.5
12/ 2 Orthogonal Operator (optional) 7.5
12/ 9 Exercise 7.5 PDF PPT
12/ 9 ~~~~~~ HW4 Due! ~~~~~~ Go to...
12/ 9 ~~~~~~ HW5 Released! ~~~~~~ Go to...
12/ 9 symmetric matrices: eigenvalues are always real (2x2 matrices) 7.6
12/ 9 thm. symmetric matrices: eigenvalues are always real (general cases) 7.6
12/ 9 thm. symmetric matrices: eigenvectors for different eigenvalues are orthogonal 7.6
12/ 9 thm. symmetric matrices are diagonalizable 7.6
12/ 9 pf. symmetric matrices are diagonalizable (proof I) 7.6
12/ 9 pf. symmetric matrices are diagonalizable (proof II) 7.6
12/ 9 ex. symmetric matrices are diagonalizable (example) 7.6
12/ 9 ex. symmetric matrices are diagonalizable (example) 7.6
12/ 9 How to diagonalize symmetric matrices 7.6
12/ 9 thm. Spectral Decomposition 7.6
12/ 9 ex. Spectral Decomposition (example) 7.6
12/ 9 Singular Value Decomposition (SVD) (optional) 7.7
12/ 9 SVD v.s. Rank (optional) 7.7
12/ 9 SVD - Low Rank Approximation (optional) 7.7
12/ 9 SVD - Application (optional) 7.7
12/ 9 SVD - proof I (optional) 7.7
12/ 9 SVD - proof II (optional) 7.7
12/ 9 SVD - proof III (optional) 7.7
Chapter 6
DueDate Topic YouTube Textbook PDF PPT
  vector space
12/ 9 原來萬物都是 vector ! 6.1
12/ 9 def. Vector Space 6.1
12/ 9 Revisit Subspace 6.1
12/ 9 Revisit Linear Combination and Span 6.2
12/ 9 ~~~~~~ HW6 Released! ~~~~~~ Go to...
12/16 Revisit Linear Transformation 6.2
12/16 Isomorphism 6.2
12/16 Revisit Basis 6.3
12/16 Vector Representation of Object 6.4
12/16 Matrix Representation of Linear Operator 6.4
12/16 Revisit Eigenvalue and Eigenvector 6.4
12/16 def. Inner Product 6.5
12/16 ex. Example 6.5
12/16 Revisit Orthogonal/Orthonormal Basis 6.5
12/16 ex. Example 6.5
12/30 ~~~~~~ HW5 Due! ~~~~~~ Go to...
12/30 ~~~~~~ HW6 Due! ~~~~~~ Go to...
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Homework

# Date Topic TA Slides Video
HW0 Colab 邱品誠
HW1 9/23~10/ 7 Cycle Detection 邱品誠
HW2 10/ 7~11/ 4 Hill Cipher 樊 樺
HW3 11/ 4~11/18 Cosine Transform and Its Application 蕭淇元
HW4 11/18~12/ 2 PageRank 邱品誠
HW5 12/ 2~12/30 Linear Regression 樊 樺
HW6 12/ 9~12/30 SVD for Image Compression 蕭淇元
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