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.
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Linear-Algebra
13 notes
Appendices
Reference material: set theory, proof techniques, real and complex numbers, polynomial algebra, calculus review, numerical computation, notation, historical notes, glossary, and index.
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.
IX. Applications
Linear regression, optimization, graphs, Markov chains, differential equations, Fourier transforms, signal processing, computer graphics, robotics, quantum mechanics, machine learning, PCA, and more.
VIII. Advanced Structures
Tensor products, exterior and symmetric algebras, multilinear maps, bilinear forms, Clifford algebras, Lie algebras, representation theory, and infinite-dimensional spaces.
VII. Numerical Linear Algebra
Floating point arithmetic, conditioning, stability, iterative solvers, Jacobi, Gauss-Seidel, conjugate gradient, Krylov subspaces, QR algorithm, sparse and randomized methods.
VI. Matrix Decompositions
LU, PLU, Cholesky, QR, Schur, SVD, polar, Hessenberg, tridiagonalization, and canonical matrix forms.
V. Eigenvalues and Spectral Theory
Eigenvalues, eigenvectors, diagonalization, the spectral theorem, Jordan canonical form, Cayley-Hamilton, matrix functions, and Perron-Frobenius theory.
IV. Inner Product Spaces
Inner products, norms, orthogonality, Gram-Schmidt, orthogonal projections, least squares, QR factorization, Hermitian spaces, and quadratic forms.
III. Linear Transformations
Linear maps, kernel and image, matrix representation, isomorphisms, projections, reflections, rotations, similarity, and invariant subspaces.
II. Vector Spaces
Abstract vector spaces, subspaces, span, linear independence, basis, dimension, coordinate systems, dual spaces, and direct sums.
I. Foundations
Scalars, vectors, matrices, linear equations, Gaussian elimination, determinants, and matrix factorizations — the bedrock of linear algebra.
15. Linear and Multilinear Algebra; Matrix Theory
This volume develops vector spaces, linear maps, matrices, and multilinear structures.