Linear Algebra/Abstract Algebra/Matrix Theory/Representation Theory

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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

Example of Linear Combination (Visual)

Linear Transformations on R^2

Examples of Linear Maps

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 Basis for a Span

Example of Linear Independence Using Determinant

Example of Kernel and Range of Linear Transformation

Linear Transformations: One-One

Linear Transformations: Onto

Example of Change of Basis

Eigenvalues and Eigenvectors

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)

Vector Length in R^n

The Standard Inner Product on R^n

Example of Fourier's Trick

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

Mystery Division Problem

GT1. Definition of Group

GT1.1. Example of Group Inverse

Order 2 Elements in Finite Group

Example of Group Cancellation Law

GT2. Definition of Subgroup

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

GT7. The Commutator Subgroup

GT8. Group Homomorphisms

GT9. Group Isomorphisms

GT10. Examples of Non-Isomorphic Groups

GT11. Group Automorphisms

GT11.1. Automorphisms of A4

GT12. Aut(Z/n) and Fermat/'s Little Theorem

GT12.1. Automorphisms of Dihedral Groups

GT13. Groups of Order 8

GT14. Semidirect Products

GT15. Group Actions

GT16. Cayley's Theorem

GT16.1 Examples of Cayley's Theorem

GT17. Symmetric and Alternating Groups

GT17.1. Permutation Matrices

GT18. Conjugacy and The Class Equation

GT18.1. Class Equation for Dihedral Groups

GT18.2. A_n is Simple (n ge 5)

GT19. Cauchy's Theorem

GT20. Overview of Sylow Theory

GT20.1. Sylow Theorems - Proofs

GT20.2 Sylow Theory for Simple 60

GT21. Internal Products

GT22. The Fundamental Theorem of Finite Abelian Groups

GT23. Composition and Classification

Example of Group Isomorphism

|Z(G)| for |G|=pq

Order 12 Subgroups in S5

Example of Conjugacy Class

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)

Example of Integral Domain

Example of Ring isomorphism 1

Order of a Finite Field

Galois Group of Order 8

Discriminant of a Cubic