Linear Algebra

Part I. Foundations

Chapter	Title
1	What Linear Algebra Is
2	Scalars, Vectors, and Fields
3	Geometry of Euclidean Space
4	Linear Equations
5	Systems of Linear Equations
6	Matrices
7	Matrix Operations
8	Elementary Row Operations
9	Gaussian Elimination
10	Gauss-Jordan Elimination
11	Invertible Matrices
12	Rank and Nullity
13	Determinants
14	Cofactors and Adjugates
15	Block Matrices
16	Matrix Factorizations Overview

Part II. Vector Spaces

Chapter	Title
17	Vector Spaces
18	Subspaces
19	Span and Linear Combination
20	Linear Independence
21	Basis
22	Dimension
23	Coordinate Systems
24	Change of Basis
25	Row Space and Column Space
26	Null Space
27	Quotient Spaces
28	Dual Spaces
29	Annihilators
30	Direct Sums
31	Affine Spaces

Part III. Linear Transformations

Chapter	Title
32	Linear Transformations
33	Kernel and Image
34	Matrix Representation of Linear Maps
35	Composition of Transformations
36	Isomorphisms
37	Automorphisms
38	Projection Operators
39	Reflection Operators
40	Rotation Operators
41	Shears and Scalings
42	Similarity Transformations
43	Invariant Subspaces
44	Cyclic Subspaces

Part IV. Inner Product Spaces

Chapter	Title
45	Inner Products
46	Norms and Metrics
47	Orthogonality
48	Orthogonal Complements
49	Orthonormal Bases
50	Gram-Schmidt Orthogonalization
51	Orthogonal Projections
52	Least Squares Problems
53	QR Factorization
54	Unitary and Orthogonal Matrices
55	Hermitian Spaces
56	Positive Definite Matrices
57	Quadratic Forms
58	Sylvester’s Law of Inertia

Part V. Eigenvalues and Spectral Theory

Chapter	Title
59	Eigenvalues
60	Eigenvectors
61	Characteristic Polynomial
62	Eigenspaces
63	Diagonalization
64	Spectral Theorem
65	Symmetric Matrices
66	Hermitian Operators
67	Normal Operators
68	Jordan Canonical Form
69	Minimal Polynomial
70	Cayley-Hamilton Theorem
71	Rational Canonical Form
72	Matrix Functions
73	Matrix Exponential
74	Perron-Frobenius Theory

Part VI. Matrix Decompositions

Chapter	Title
75	LU Decomposition
76	PLU Decomposition
77	Cholesky Decomposition
78	QR Decomposition
79	Schur Decomposition
80	Singular Value Decomposition
81	Polar Decomposition
82	Hessenberg Form
83	Tridiagonalization
84	Canonical Matrix Forms

Part VII. Numerical Linear Algebra

Chapter	Title
85	Floating Point Arithmetic
86	Conditioning and Stability
87	Error Analysis
88	Iterative Methods for Linear Systems
89	Jacobi Method
90	Gauss-Seidel Method
91	Conjugate Gradient Method
92	Krylov Subspaces
93	Power Iteration
94	QR Algorithm
95	Sparse Matrices
96	Structured Matrices
97	Randomized Linear Algebra

Part VIII. Advanced Structures

Chapter	Title
98	Tensor Products
99	Exterior Algebra
100	Symmetric Algebra
101	Multilinear Maps
102	Bilinear Forms
103	Alternating Forms
104	Clifford Algebras
105	Lie Algebras
106	Representation Theory Basics
107	Infinite-Dimensional Vector Spaces
108	Functional Analysis Connections

Part IX. Applications

Chapter	Title
109	Linear Regression
110	Optimization and Linear Algebra
111	Graphs and Adjacency Matrices
112	Markov Chains
113	Differential Equations
114	Fourier Series and Transforms
115	Signal Processing
116	Computer Graphics
117	Robotics and Kinematics
118	Control Theory
119	Quantum Mechanics
120	Machine Learning
121	Principal Component Analysis
122	PageRank and Network Analysis
123	Coding Theory
124	Cryptography
125	Finite Element Methods
126	Scientific Computing

Part X. Specialized Topics

Chapter	Title
127	Complex Vector Spaces
128	Finite Fields
129	Linear Algebra over Arbitrary Fields
130	Modules and Linear Algebra
131	Category-Theoretic Perspective
132	Convex Geometry
133	Random Matrices
134	Numerical Optimization
135	Operator Theory
136	Spectral Graph Theory
137	Compressed Sensing
138	Tensor Decompositions
139	Geometric Algebra
140	Modern Applications in AI

Appendices

Appendix	Title
A	Set Theory and Logic
B	Proof Techniques
C	Real and Complex Numbers
D	Polynomial Algebra
E	Calculus Review
F	Numerical Computation
G	Mathematical Notation
H	Historical Notes
I	Glossary
J	Theorem Index
K	Symbol Index

This structure combines the standard undergraduate sequence with numerical, computational, geometric, and modern applied directions. It follows the progression used in many major references and courses: systems and matrices, vector spaces, linear maps, inner products, spectral theory, decompositions, then advanced and applied topics.

I. FoundationsScalars, vectors, matrices, linear equations, Gaussian elimination, determinants, and matrix factorizations — the bedrock of linear algebra.

16 pages · 144 min

II. Vector SpacesAbstract vector spaces, subspaces, span, linear independence, basis, dimension, coordinate systems, dual spaces, and direct sums.

15 pages · 130 min

III. Linear TransformationsLinear maps, kernel and image, matrix representation, isomorphisms, projections, reflections, rotations, similarity, and invariant subspaces.

13 pages · 136 min

IV. Inner Product SpacesInner products, norms, orthogonality, Gram-Schmidt, orthogonal projections, least squares, QR factorization, Hermitian spaces, and quadratic forms.

14 pages · 131 min

V. Eigenvalues and Spectral TheoryEigenvalues, eigenvectors, diagonalization, the spectral theorem, Jordan canonical form, Cayley-Hamilton, matrix functions, and Perron-Frobenius theory.

16 pages · 158 min

VI. Matrix DecompositionsLU, PLU, Cholesky, QR, Schur, SVD, polar, Hessenberg, tridiagonalization, and canonical matrix forms.

10 pages · 108 min