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Linear Algebra
00 Major Problems in Applied Linear Algebra
0 The Applied Math Modeling Problem
00 Major Problems in Applied Linear Algebra
1 The Matrix-Vector Multiplication Problem
2A The Nonsingular Linear-Systems Problem
2B The General Linear-Systems Problem
3 The Full-Rank Least-Squares Problem
4 The Standard Eigenvalue Problem
The Three Types of Solutions to General Linear Systems
What do quadratic equations teach us about general linear systems
01 Set Theory
01 Set Theory
1.1 Set
1.10 Proving Set Equality
1.11 Empty Set
1.12 Classic Subset Results Theorem
1.2 Element Enumeration
1.3 Element Inclusion and Truth Values
1.4 Stanley's Brace Notation & Ellipses
1.5 Set Builder Notation
1.6 Important Number Systems
1.7 Subset
1.8 Proving Subsets
1.9 Set Equality
02 Relations and Functions
02 Relations and Functions
2.1 Cross Product of Sets
2.2 Examples of Cartesian Products
2.3 Relation
2.3.1 Traffic Light Relation
2.3.2 Dial Pad Relation
2.4 Domain, Range, & More
2.4.1 Ellipse Relation Domain & Range
2.5 Function (Set Theory)
2.6 Important Sets of Functions
Fundamental Theorems of Calculus
03 Vector Modeling
03 Vector Modeling
3.1 Column Vector
3.1.1 Vertex Model of Points
3.1.2 Vector Model of Mass-Spring Chain
3.2 The ith Coefficient of a Column Vector
3.3 R^n
3.4 Row Vector
3.5 Equal Column Vectors
04 Vector Arithmetic
04 Vector Arithmetic
4.1 Scalar-Vector Multiplication
4.1.1 Newton's Second Law & Spring Forces
4.1.2 Geometry of Scalar-Vector Multiplication
4.2 Column Vector Addition
4.2.1 Shifting a Vertex Model
4.2.2 Vector Arithmetic in Mass Spring Chain
4.2.3 Hooke's Law Data Analysis
4.3 Algebraic Properties of Vector Arithmetic
4.4 Transpose of a Vector
4.5 Algebraic Properties of Vector Transposes
05 Inner Products and Vector Norms
05 Inner Products and Vector Norms
5.1 Inner Product Between Vectors
5.1.1 Final grade calculations using inner product
5.2 Algebraic Properties of Inner Products
5.3 The 2-norm of a Vector
5.4 Algebraic Properties of the 2-norm of a Vector
5.5 Pythagorean Theorem
5.6 Law of Cosines
5.7 Cosine Formula for Inner Product
5.8 Cauchy-Schwarz Inequality
5.9 Orthogonal Vectors
06 Span and Linear Independence
06 Span and Linear Independence
6.1 Linear Combination of Vectors
6.1.1 Creating Linear Combinations
6.1.2 Steph Curry Model
6.2 Linearity
6.3 Span of a set of vectors
6.4 Linearly Dependent Vectors
6.4.1 Span of 3 Vectors Example
6.4.2 Examples of Linear Dependence
6.5 Theorem of Vector List Size m x 1
6.5.1 Example of Linear Dependent Set
6.6 Test for Linear Dependence
6.6.1 Using Test for Linear Dependence
6.7 Linearly Independent Vectors
6.7.1 Fantasy Basis (Elementary Standard Basis for R5
6.8 Test for Linear Independence
07 Matrix Modeling
07 Matrix Modeling
7.1 Entry-by-Entry Definition of Matrix
7.2 Dimensions of a Matrix
7.3 Examples of Matrix Models
7.4 Undirected Graphs
7.5 Directed Graphs
7.6 Equal Matrices
08 Anatomy of Matrices
08 Anatomy of Matrices
8.1 Matrix Anatomy
8.11 Colon Notation
8.12 Column Operator
8.13 Column Partition of a Matrix
8.14 Row Partition of a Matrix
8.15 Column and Row Partition of Identity Matrix
8.2 Entries of a Matrix
8.3 Sparse Matrices
8.4 Special Sparsity Notation
8.5 Diagonal Matrix
8.6 Identity Matrix
8.7 Lower-Triangular Matrix
8.8 Unit Lower-Triangular Matrix
8.9 Upper-Triangular Matrix
09 Matrix Arithmetic
09 Matrix Arithmetic
9.1 Outer Product of Vectors
9.10 Transposition Matrix
9.10.1 Transposition Matrix as Outer Product
9.11 Givens Rotation
9.12 Gauss Transform
9.13 Transpose of a Matrix
9.14 Algebraic Properties of Matrix Transposes
9.2 Example of Matrix Units
9.3 Example of Outer Product
9.4 Matrix-Matrix Addition
9.4.1 Identity as Matrix Matrix Addition Example
9.5 Scalar-Matrix Multiplication
9.5.1 Matrix as Linear Combination of Outer Products
9.6 Algebraic Properties of Matrix Operations
9.7 Rank-one Update
9.8 Shear Matrix
9.9 Dilation Matrix
Lesson 9 Full Solutions
10 Matrix-Vector Multiplication
10 Matrix-Vector Multiplication
10.1 Matrix-Column-Vector Multiplication via Linear Combinations
10.1.1 Example of Matrix-Column-Vector Multiplication via Linear Combinations
10.2 Matrix-Column-Vector Multiplication via Dot Products
10.2.1 Example of Matrix-Column-Vector Multiplication Using Dot Products
10.3 Properties of Matrix-Column-Vector Multiplication
10.4 Row-Vector-Matrix Multiplication via Linear Combinations
10.4.1 Example of Row Vector Matrix Multiplication via Linear Combinations
10.5 Row-Vector-Matrix Multiplication via Dot Products
10.5.1 Example of Row-Vector-Matrix Multiplication via Dot Products
10.6 Relation Between Matrix-Vector Multiplication and Matrix Partitions
10.7 Properties of Row-Vector-Matrix Multiplication
10.8 Transpose of a Matrix-Vector Product
11 Matrix-Matrix Multiplication
11 Matrix-Matrix Multiplication
11.1 Anatomy of Matrix-Matrix Multiplication
11.2 Matrix-Matrix Multiplication via Linear Combination of Columns
11.2.1 Example of Matrix-Matrix Multiplication via Linear Combination of Columns
11.3 Matrix-Matrix Multiplication via Linear Combination of Rows
11.3.1 Example of Matrix-Matrix Multiplication via Linear Combination of Rows
11.4 Matrix-Matrix Multiplication via Dot Products
11.4.1 Example of Matrix-Matrix Multiplication via Dot Products
11.5 Matrix-Matrix Multiplication via Outer Products
11.6 Algebraic Properties of Matrix-Matrix Multiplication
11.7 Applications for Mass Spring Chain
12 Nonsingular Linear Systems
12 Nonsingular Linear Systems
12.1 The Square Linear-Systems Problem
12.2 Backwards Substitution
12.2.1 Nonsingular Matrix to Model Gravity
12.3 Regular Matrix
12.4 Foundation to Solve NLSP
12.5 Forward Substitution
12.6 Coefficient Matrix of a Linear System
12.7 Fundamental Questions about Linear Systems
12.8 Solution Set to Square Linear-Systems Problem
12.9 Determine Solution Validity for NLSP
13 Invertible Matrices
13 Invertible Matrices
13.1 Inverse of a Square Matrix
13.2 Inverses of Elementary Matrices
13.3 Cramer's Rule for Inverse of a 2x2 System
13.4 Properties of Matrix Inverses
14 Invertible Matrix Theorem
14 Invertible Matrix Theorem
14.1 The Invertible Matrix Theorem Part 1
14.2 The Invertible Matrix Theorem Part 2
14.3 The Invertible Matrix Theorem Part 3
15 LU Factorization without Pivoting
15 LU Factorization without Pivoting
15.1 LU Factorization without Pivoting
15.2 Review of Nonsingular Gravity Model
15.3 4-by-4 LU Factorization
16 Determinants
16 Determinants
16.1 Permutation
16.1.1 Permutation of a Set A
16.2 Symmetric Groups
16.2.1 Inversion of a pair (i, j) with respect to pi
16.2.2 Set of All Inversions for a Given pi in Sn
16.2.3 Sign of a Permutation
16.2.4 Cycles
16.3 Transpositions Generate Permutations
16.4 Deriving the Determinant Function
16.5 Permutation Definition of Determinant
16.6 Rule of Sarrus
16.7 Properties of Determinants
17 General Linear-Systems
16.2 Row Echelon Form
17 General Linear-Systems
17.1 The General Linear-Systems Problem
17.2 Row Echelon Form
17.3 Reduced Row Echelon Form
17.4 Model of Airplane Descent Path to Landing
17.5 Solving GLSP for Airplane Descent Path
17.6 Toy GLSP
18 Solution Sets for General-Linear Systems
18 Solution Sets for General-Linear Systems
18.1 Template for Complete Solutions to GLSP
18.10 Rank of a Matrix
18.2 Example of Template for Complete Solutions
18.3 Homogenous Linear System
18.4 Solution to HSLP using RREF
18.5 Solution to HLSP
18.6 Superposition of Solutions for HLSP
18.7 Superposition for Solution to GLSP
18.8 Existence of Solution to GLSP
18.9 Algorithm to Solve GLSP Using RREF
27 Standard Eigenvalue Problem
27 Standard Eigenvalue Problem
27.1 The Standard Eigenvalue Problem
27.10 Solving the Coupled Pendula SEP
27.2 Intro to Coupled Pendula Problem
27.3 McCusker Apparatus
27.4 The Coupled Pendula Problem
27.5 Motion of a Single Pendulum
27.6 Linearize the Nonlinear ODE for a Simple Pendulum
27.7 Mathematize the Coupled Pendula Problem
27.8 State Coupled Pendula ODEs Using Matrices
27.9 The SEP to Model Coupled Pendula
28 Eigenvalue Theory
28 Eigenvalue Theory
28.1 Case Studies of Eigenvalues of 2x2 Matrices Analysis, Categories, and Relations
28.2 Positive Definite Matrix
28.3 How to Identify if 2x2 Matrix is Positive Definite
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Lesson 9 Full Solutions
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