Linear Algebra

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Symbol Description
Biography
1. Matrices
	1.1. Real and Complex Matrices
	1.2. Matrix Calculus
	1.3. Matrix Inverses
	1.4. Elementary Matrices
		1.4.1. LU and LDU factorisations
	1.5. Exercises
	1.6. At a Glance
2. Determinant
	2.1. Axiomatic Definition
	2.2. Leibniz’s Formula
	2.3. Laplace’s Formula
	2.4. Exercises
	2.5. At a Glance
3. Vector Spaces
	3.1. Vector Spaces
	3.2. Linear Independence
	3.3. Bases and Dimension
		3.3.1. Matrix spaces and spaces of polynomials
		3.3.2. Existence and construction of bases
	3.4. Null Space, Row Space, and Column Space
		3.4.1. Ax = b
	3.5. Sum and Intersection of Subspaces
	3.6. Change of Basis
	3.7. Exercises
	3.8. At a Glance
4. Eigenvalues and Eigenvectors
	4.1. Spectrum of a Matrix
	4.2. Spectral Properties
	4.3. Similarity and Diagonalisation
	4.4. Jordan Canonical Form
		4.4.1. Nilpotent matrices
		4.4.2. Generalised eigenvectors
		4.4.3. Jordan canonical form
	4.5. Exercises
	4.6. At a Glance
5. Linear Transformations
	5.1. Linear Transformations
	5.2. Matrix Representations
	5.3. Null Space and Image
		5.3.1. Linear transformations T : Kn → Kk
		5.3.2. Linear transformations T : U → V
	5.4. Isomorphisms and Rank-nullity Theorem
	5.5. Composition and Invertibility
	5.6. Change of Basis
	5.7. Spectrum and Diagonalisation
	5.8. Exercises
	5.9. At a Glance
6. Inner Product Spaces
	6.1. Real Inner Product Spaces
	6.2. Complex Inner Product Spaces
	6.3. Orthogonal Sets
		6.3.1. Orthogonal complement
		6.3.2. Orthogonal projections
		6.3.3. Gram–Schmidt process
	6.4. Orthogonal and Unitary Diagonalisation
	6.5. Singular Value Decomposition
	6.6. Affine Subspaces of Rn
	6.7. Exercises
	6.8. At a Glance
7. Special Matrices by Example
	7.1. Least Squares Solutions
	7.2. Markov Chains
		7.2.1. Google matrix and PageRank
	7.3. Population Dynamics
	7.4. Graphs
	7.5. Differential Equations
	7.6. Exercises
	7.7. At a Glance
8. Appendix
	8.1. Uniqueness of Reduced Row Echelon Form
	8.2. Uniqueness of Determinant
	8.3. Direct Sum of Subspaces
9. Solutions
	9.1. Solutions to Chapter 1
	9.2. Solutions to Chapter 2
	9.3. Solutions to Chapter 3
	9.4. Solutions to Chapter 4
	9.5. Solutions to Chapter 5
	9.6. Solutions to Chapter 6
	9.7. Solutions to Chapter 7
Bibliography
Index