Linear Algebra/Abstract Algebra/Matrix Theory/Representation Theory
Linear Algebra
Row Reduction for a System of Two Linear Equations
Solving a SLE in 3 Variables with Row Operations 1
Solving a SLE in 3 Variables with Row Operations 2
Solving a 2x2 SLE Using a Matrix Inverse
Consistency of a System of Linear Equations
Inverse of 3 x 3 Matrix Using Row Operations 1
Inverse of 3 x 3 Matrix Using Row Operations 2
Inverse of 4 x4 Matrix Using Row Operations
Example of Determinant Using Row Echelon Form
Inverse of 3 x 3 Matrix Using Adjugate Formula
Inverse of 4 x 4 Matrix Using Adjugate Formula
Cramer's Rule for Three Linear Equations
Determinant of a 4 x 4 Matrix Using Cofactors
Determinant of a 4 x 4 Matrix Using Row Operations
Example of Linear Combination (Visual)
Evaluating Linear Transformations Using a Basis
Evaluating Linear Transformations Using a Basis 2
Example of Checking for Basis Property
Example of Basis for a Null Space
Example of Linear Independence Using Determinant
Example of Kernel and Range of Linear Transformation
Linear Transformations: One-One
Example of Eigenvector: Markov Chain
Example of Diagonalizing a 2 x 2 Matrix
Example of Power Formula for a Matrix
The Fibonacci Numbers Using Linear Algebra (HD Version)
The Standard Inner Product on R^n
Example of Orthogonal Subspace
Orthogonal Transformations 1: 2x2 Case
Orthogonal Transformations 2: 3x3 Case
Example of Orthogonal Projection
Example of Gram-Schmidt Orthogonalization
QR-Decomposition for a 2x2 Matrix
Beyond Eigenspaces: Real Invariant Planes
Spectral Theorem for Real Matrices: General 2x2 Case
Spectral Theorem for Real Matrices: General nxn Case
Example of Spectral Theorem (2x2 Symmetric Matrix)
Example of Spectral Theorem (3x3 Symmetric Matrix)
Example of Spectral Decomposition
Beyond Eigenspaces 2: Complex Form
Abstract Algebra
Group Theory
GT1.1. Example of Group Inverse
Order 2 Elements in Finite Group
Example of Group Cancellation Law
GT3. Cosets and Lagrange\'s Theorem
GT4. Normal Subgroups and Quotient Groups
GT5. Index 2 Theorem and Dihedral Groups
GT6. Centralizers, Normalizers, and Direct Products
GT10. Examples of Non-Isomorphic Groups
GT12. Aut(Z/n) and Fermat/'s Little Theorem
GT12.1. Automorphisms of Dihedral Groups
GT16.1 Examples of Cayley's Theorem
GT17. Symmetric and Alternating Groups
GT18. Conjugacy and The Class Equation
GT18.1. Class Equation for Dihedral Groups
GT18.2. A_n is Simple (n ge 5)
GT20. Overview of Sylow Theory
GT20.1. Sylow Theorems - Proofs
GT20.2 Sylow Theory for Simple 60
GT22. The Fundamental Theorem of Finite Abelian Groups
GT23. Composition and Classification
Sylow Theory for Order 12 Groups 1
Sylow Theory for Order 12 Groups 2
Simple Group 168 - Sylow Theory - Part 1
Simple Group 168 - Sylow Theory - Part 2
Example of Group: GL(2, R) (1 of 3)
Example of Group: GL(2,R) (2 of 3)
Example of Group: GL(2, R) (3 of 3)
Ring Theory
RNT1.2. Definition of an Integral Domain
RNT1.2.1. Example of an Integral Domain
RNT1.2.2. Order of a Finite Field
RNT1.4. Ideals and Quotient Rings
RNT1.4.1. Example of Quotient Ring
RNT2.1. Maximal Ideals and Fields
RNT2.1.1. Finite Fields ofOrders 4 and 8
RNT2.2. Principal Ideal Domains
RNT2.3.1. Euclidean Algortihm for Gaussian Integers
RNT2.5. Polynomial Rings over Fields
RNT2.5.1. Euclidean Algorithm for Z/3[x]
RNT2.6.2. Eisenstein's Criterion
Field Theory
FIT2.2.1. Example: Cubic Extension
FIT2.2.2. Example: Quartic Extension
FIT2.3.2. Cardinality and Transcendentals
FIT2.3.3. Algebraic Extensions
FIT3.1.1. Roots of Polynomials
FIT3.1.2. Roots of Real Polynomials
FIT3.1.3. Example of Splitting Field
FIT3.1.4. Factoring Example: Artin-Schreier Polynomials
FIT3.2.1. Cyclotomic Polynomials
FIT3.2.2. Mobius Inversion Formula
FIT4.1. Galois Group of a Polynomial
FIT4.2. Automorphisms and Degree
FIT4.3. Galois Correspondence 1 - Examples
FIT4.3.1. Galois Group of Order 8
Matrix Theory
Matrix Inverse over the Complex Numbers
Cramer's Rule over the Complex Numbers
Positive Semi-Definite Matrix 1: Square Root
Positive Semi-Definite Matrix 2: Spectral Theorem
Positive Semi-Definite Matrix 3: Factorization of Invertible Matrices
Example of Skew-Symmetric Matrix
Example of Invariant Subspace 1
Cayley-Hamilton Theorem for 2x2 Matrices
Inverse of a Matrix Using the Cayley-Hamilton Theorem
Cayley-Hamilton Theorem: General Case
Cayley-Hamilton Theorem: Example 1
Cayley-Hamilton Theorem Example 2
Overview of Minimal Polynomials
Minimal Polynomials and Diagonal Form
Example of Simultaneous Diagonalization
Overview of Jordan Canonical Form
Example of Jordan Canonical Form: 2x2 Case
Example of Jordan Canonical Form: General Properties
Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 1
Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 2
Example of Rational Canonical Form 1: Single Block
Example of Rational Canonical Form 2: Several Blocks
Example of Rational Canonical Form 3
Exponential of 2x2 Matrix 1: Complex Case
Exponential of 2x2 Matrix 2: Traceless Case
The Fibonacci Numbers Using Linear Algebra
Linear Algebra: The Binet Formula - Part 2
Example of Group Automorphism 1 (Requires Linear Algebra)
Example of Group Automorphism 2: G = Z/4 x Z/4 (Requires Linear Algebra)
Group Theory: The Simple Group of Order 168 - Part 1
Group Theory: The Simple Group of Order 168 - Part 2
Representation Theory of Finite Groups
RT4.1: Constructions from Linear Algebra
RT4.1.1: Complex Conjugate Representations
RT6: Representations on Function Spaces
RT7.1: Finite Abelian Groups 1: Character Orthogonality
RT7.2: Finite Abelian Groups 2: Fourier Analysis
RT7.3: Finite Abelian Groups 3: Convolution
RT8.1: Finite Groups 1: Schur Orthogonality Relations
RT8.2: Finite Groups 2: Classification of Irreducibles
RT8.3: Finite Groups 3: Projection to Irreducibles
RT9.1: Application: Normal Modes
Character Tables for S4 and A4
Character Tables for S5 and A5
Basic Algebraic Geometry and Toric Varieties
BAG1.1. Toric Varieties 1 - Affine Varieties over C
BAG1.2. Toric Varieties 2 - Affine Toric Varieties
BAG1.3. Yoric Varieties 3 - Coordinate Rings and Morphisms
BAG1.4. Toric Varieties 4 - Spec(R) and Affine Semigroups
BAG1.5. Toric Varieties 5 - Polyhedral Cones afor Affine Toric Varieties
BAG1.6. Toric Varieties 6 - Faces and Localization
BAG1.7. Toric Varieties 7 - Overview of Smoothness and Normality
BAG2.1. Projective Toric Varieties - Part 1