A Course in Linear Algebra

✍ Scribed by Raju K. George, Abhijith Ajayakumar

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 555
• Category: Library

✦ Synopsis

Designed for senior undergraduate and graduate courses in mathematics and engineering, this self-contained textbook discusses key topics in linear algebra with real-life applications. Split into two parts—theory in part I and solved problems in part II—the book makes both theoretical and applied linear algebra easily accessible. Topics such as sets and functions, vector spaces, linear transformations, eigenvalues and eigenvectors, normed spaces, and inner product spaces are discussed in part I; while in part II, over 500 meticulously solved problems show how to use linear algebra in real-life situations. A must-have book for linear algebra courses; it also serves as valuable supplementary material.

✦ Table of Contents

Preface
Contents
Part I Theory of Linear Algebra
1 Preliminaries
1.1 Sets and Functions
1.2 Metric Spaces
1.3 Some Important Algebraic Structures
1.4 Polynomials
1.5 Matrices
1.6 Euclidean Space double struck upper R Superscript nmathbbRn
1.7 System of Linear Equations
1.8 Exercises
2 Vector Spaces
2.1 Introduction
2.2 Subspaces
2.3 Linear Dependence and Independence
2.4 Basis and Dimension
2.5 Sum and Direct Sum
2.6 Exercises
3 Linear Transformations
3.1 Introduction
3.2 Range Space and Null Space
3.3 Matrix Representation of a Linear Transformation
3.4 Algebra of Linear Transformations
3.5 Invertible Linear Transformations
3.6 Change of Coordinate Matrix
3.7 Linear Functionals and Dual Space
3.8 Exercises
4 Eigenvalues and Eigenvectors
4.1 Eigenvalues and Eigenvectors
4.2 Diagonalization
4.3 Schur Triangularization Theorem
4.4 Generalized Eigenvectors
4.5 Jordan Canonical Form
4.6 Exercises
5 Normed Spaces and Inner Product Spaces
5.1 Normed Linear Spaces
5.2 Inner Product Spaces
5.3 Orthogonality of Vectors and Orthonormal Sets
5.4 Orthogonal Complement and Projection
5.5 Exercises
6 Bounded Linear Maps
6.1 Bounded Linear Maps
6.2 Adjoint of a Bounded Linear Map
6.3 Self-adjoint Operators
6.4 Normal, Unitary Operators
6.5 Singular Value Decomposition
6.6 Generalized Inverse of a Matrix
6.7 Iterative Methods for System of Linear Equations
6.8 Exercises
7 Applications
7.1 Applications Involving System of Equations
7.2 Cryptography
7.3 Markov Process
7.4 Coupled Harmonic Oscillators
7.5 Satellite Control Problem
7.6 Artificial Neural Network as Linear Regression Model
Part II Solved Problems
8 Solved Problems—Preliminaries
9 Solved Problems—Vector Spaces
10 Solved Problems—Linear Transformations
11 Solved Problems—Eigenvalues and Eigenvectors
12 Solved Problems—Normed Spaces and Inner Product Spaces
13 Solved Problems—Bounded Linear Maps
Appendix Appendix
A.1 Determinants
A.2 Fourier Series
References
Index