Linear Algebra

A comprehensive book covering linear algebra from foundations through spectral theory, matrix decompositions, numerical methods, and modern applications in ten parts with appendices.

11 items

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.

Writes › Book › Linear Algebra ›

I. Foundations

Scalars, vectors, matrices, linear equations, Gaussian elimination, determinants, and matrix factorizations — the bedrock of linear algebra.
linear-algebra mathematics book
Writes › Book › Linear Algebra ›

II. Vector Spaces

Abstract vector spaces, subspaces, span, linear independence, basis, dimension, coordinate systems, dual spaces, and direct sums.
linear-algebra mathematics book
Writes › Book › Linear Algebra ›

III. Linear Transformations

Linear maps, kernel and image, matrix representation, isomorphisms, projections, reflections, rotations, similarity, and invariant subspaces.
linear-algebra mathematics book
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IV. Inner Product Spaces

Inner products, norms, orthogonality, Gram-Schmidt, orthogonal projections, least squares, QR factorization, Hermitian spaces, and quadratic forms.
linear-algebra mathematics book
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V. Eigenvalues and Spectral Theory

Eigenvalues, eigenvectors, diagonalization, the spectral theorem, Jordan canonical form, Cayley-Hamilton, matrix functions, and Perron-Frobenius theory.
linear-algebra mathematics book
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VI. Matrix Decompositions

LU, PLU, Cholesky, QR, Schur, SVD, polar, Hessenberg, tridiagonalization, and canonical matrix forms.
linear-algebra mathematics book
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VII. Numerical Linear Algebra

Floating point arithmetic, conditioning, stability, iterative solvers, Jacobi, Gauss-Seidel, conjugate gradient, Krylov subspaces, QR algorithm, sparse and randomized methods.
linear-algebra mathematics book
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VIII. Advanced Structures

Tensor products, exterior and symmetric algebras, multilinear maps, bilinear forms, Clifford algebras, Lie algebras, representation theory, and infinite-dimensional spaces.
linear-algebra mathematics book
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IX. Applications

Linear regression, optimization, graphs, Markov chains, differential equations, Fourier transforms, signal processing, computer graphics, robotics, quantum mechanics, machine learning, PCA, and more.
linear-algebra mathematics book
Writes › Book › Linear Algebra ›

X. Specialized Topics

Complex vector spaces, finite fields, modules, category theory, convex geometry, random matrices, operator theory, spectral graph theory, compressed sensing, tensor decompositions, geometric algebra, and AI applications.
linear-algebra mathematics book
Writes › Book › Linear Algebra ›

Appendices

Reference material: set theory, proof techniques, real and complex numbers, polynomial algebra, calculus review, numerical computation, notation, historical notes, glossary, and index.
linear-algebra mathematics book