# 📘 Linear Algebra – Lecture Series
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Lecture 1: Matrix Rank and Elementary Transformations
- 1.1 Matrix Rank
- 1.2 Row Reduced Echelon Form
- 1.3 Elementary Transformations
- 1.4 Elementary Matrices
- 1.5 Normal Form Reduction
Lecture 2: Matrix Equivalence and Linear Systems
- 2.1 Matrix Equivalence
- 2.2 Consistency of Linear Systems
Lecture 3: Introduction to Groups, Rings, and Fields
- 3.1 Algebraic Structures and Binary Operations
- 3.2 Groups
- 3.3 Subgroups
- 3.4 Group Homomorphisms
- 3.5 Rings
- 3.6 Integral Domains and Fields
- 3.7 Subrings and Ring Homomorphisms
Lecture 4: Vector Spaces — Definitions and Examples
- 4.1 Vector Spaces
Lecture 5: Vector Spaces and Subspaces
- 5.1 Vector Space of *n*-Tuples
- 5.2 Vector Subspaces
Lecture 6: Span, Linear Combinations, and Dependence
- 6.1 Linear Combinations
- 6.2 Span
- 6.3 Linear Dependence and Independence
Lecture 7: Bases and Dimensions of Vector Spaces
- 7.1 Bases
- 7.2 Dimension
Lecture 8: Subspaces, Operations, and Direct Sums
- 8.1 Subspace Operations
- 8.2 Direct Sums
- 8.3 Subspace Verification and Bases
Lecture 9: Linear Systems and Wronskian
- 9.1 Homogeneous Linear Systems
- 9.2 Linear Independence and the Wronskian
Lecture 10: Basis Reduction and Coordinate Changes
- 10.1 Reduction to Basis
- 10.2 Extension to Basis
- 10.3 Coordinate Vectors
- 10.4 Change of Basis
Lecture 11: Linear Transformations and Isomorphisms
- 11.1 Linear Transformations
- 11.2 Linear Algebraic Structures
Lecture 12: Rank, Nullity, and Kernel
- 12.1 Range and Kernel
- 12.2 Examples
Lecture 13: Matrix Representation and Functionals
- 13.1 Matrix of a Linear Transformation
- 13.2 Change of Basis
- 13.3 Composition and Inverse
- 13.4 Linear Functionals
Lecture 14: Normed and Inner Product Spaces
- 14.1 Normed Linear Spaces
- 14.2 Inner Product Spaces
Lecture 15: Properties of Inner Product Spaces
- 15.1 Cauchy-Schwarz Inequality
- 15.2 Triangle Inequality
- 15.3 Other Properties
- 15.4 Orthogonality
Lecture 16: Orthonormality and Gram-Schmidt
- 16.1 Orthonormal Sets
- 16.2 Gram-Schmidt Orthogonalization
- 16.3 Orthogonal Complements
Lecture 17: Eigenvalues and Eigenvectors
- 17.1 Characteristic Value Problem
- 17.2 Examples
Lecture 18: Cayley-Hamilton Theorem
- 18.1 Theorem Statement
- 18.2 Examples
Lecture 19: Minimal Polynomial and Multiplicity
- 19.1 Algebraic and Geometric Multiplicity
- 19.2 Minimal Polynomial
- 19.3 Examples
Lecture 20: Orthogonal and Unitary Matrices
- 20.1 Inner Product and Norm
- 20.2 Unitary and Orthogonal Matrices
- 20.3 Examples
Lecture 21: Normal Matrices and Similarity
- 21.1 Normal Matrices
- 21.2 Similarity
- 21.3 Examples
Lecture 22: Diagonalization and Triangularization
- 22.1 Diagonalization
- 22.2 Triangularization
Lecture 23: Introduction to Quadratic Forms
- 23.1 Quadratic Forms
- 23.2 Examples
Lecture 24: Canonical Forms of Quadratic Forms
- 24.1 Diagonalization of Quadratic Forms
- 24.2 Examples
Lecture 25: Applications to Conic Sections
- 25.1 Optimization
- 25.2 Conic Sections
- 25.3 Examples
Lecture 26: Introduction to Bilinear Forms
- 26.1 Bilinear Forms
- 26.2 Examples
Lecture 27: Symmetric Bilinear Forms
- 27.1 Symmetric Bilinear Forms
- 27.2 Examples
Lecture 28: Hermitian Forms
- 28.1 Complex Bilinear Forms
- 28.2 Hermitian Forms
- 28.3 Examples
Lecture 29: Jordan Normal Form
- 29.1 Jordan Block
- 29.2 Jordan String
- 29.3 Jordan Normal Form
Lecture 30: Properties and Applications of JNF
- 30.1 Similarity to Transpose
- 30.2 Polynomial Equations
- 30.3 Polynomials via JNF
- 30.4 Examples
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