Linear Algebra And Optimization With Applications To Machine Learning - Volume I

This book provides the mathematical fundamentals of linear algebra to practicers in computer vision, machine learning, robotics, applied mathematics, and electrical engineering. By only assuming a knowledge of calculus, the authors develop, in a rigorous yet down to earth manner, the mathematical theory behind concepts such as: vectors spaces, bases, linear maps, duality, Hermitian spaces, the spectral theorems, SVD, and the primary decomposition theorem. At all times, pertinent real-world applications are provided. This book includes the mathematical explanations for the tools used which we believe that is adequate for computer scientists, engineers and mathematicians who really want to do serious research and make significant contributions in their respective fields.

Contents:

• Introduction
• Vector Spaces, Bases, Linear Maps
• Matrices and Linear Maps
• Haar Bases, Haar Wavelets, Hadamard Matrices
• Direct Sums, Rank-Nullity Theorem, Affine Maps
• Determinants
• Gaussian Elimination, LU-Factorization, Cholesky Factorization, Reduced Row Echelon Form
• Vector Norms and Matrix Norms
• Iterative Methods for Solving Linear Systems
• The Dual Space and Duality
• Euclidean Spaces
• QR-Decomposition for Arbitrary Matrices
• Hermitian Spaces
• Eigenvectors and Eigenvalues
• Unit Quaternions and Rotations in SO(3)
• Spectral Theorems in Euclidean and Hermitian Spaces
• Computing Eigenvalues and Eigenvectors
• Graphs and Graph Laplacians; Basic Facts
• Spectral Graph Drawing
• Singular Value Decomposition and Polar Form
• Applications of SVD and Pseudo-Inverses
• Annihilating Polynomials and the Primary Decomposition
• Bibliography
• Index

Readership: Undergraduate and graduate students interested in mathematical fundamentals of linear algebra in computer vision, machine learning, robotics, applied mathematics, and electrical engineering.Vectors;Basis;Linear Maps;Kernels;Affine Maps;Determinants;LU;Cholesky;Rref;Haar Wavelets;Curve Interpolation;Matrix Norms;Dual Space;Euclidean Spaces;Inner Product;QR-Decomposition;Hermitian Spaces;Orthogonal Group;Matrix Exponential;Quaternions;Eigenvalues;Adjoint;Spectral Graph Theorem;Graph Laplacian;SVD;Least Squares Problem;Pseudo-Inverse;PCA (Principal Component Analysis);Primary Decomposition Theorem0Key Features:

• This book fills a gap in the market in that it is a mathematically rigorous book which provides topical applications for the fields of machine learning, computer vision, and robotics
• This book covers, in more depth than usual, duality (Chapter 9), vector and matrix norms (Chapter 7), and the spectral theorems (Chapter 15)
• This book will contain classroom tested exercises Professor Gallier has successfully used throughout his many years of teaching CIS 515: Fundamentals of Linear Algebra and Optimization. These multi-faceted exercises contain both mathematical proofs and computer programming