Elements of Linear Algebra and Matrix Theory

TItle
Preface
Contents
1. Finite-dimensional Vector Spaces
	1.1. Vector Spaces
	1.2. Directed Line Segments as Vectors
	1.3. Geometric Vectors and Coordinate Spaces
	1.4. Subspaces
	1.5. Solutions of Linear Equations: Gauss Reduction
	1.6. Solutions of Linear Equations: Determinants
	1.7. Linear Dependence of Vectors
	1.8. Basis and Dimension
	1.9. Two Important Theorems
2. Linear Transformations and Matrices
	2.1. Linear Transformations
	2.2. Properties of Linear Transformations
	2.3. Operations on Linear Transformations
	2.4. Linear Functionals
	2.5. Annihilators
	2.6. Nonsingular Linear Transformations
	2.7. Matrices of Linear Transformations
	2.8. Matrices as Multiplicative Systems
	2.9. Change of Basis
	2.10. Rank
	2.11. An Important Note of Clarification
3. Determinants and Systems of Linear Equations
	3.1. Matrices and Linear Systems
	3.2. Elementary Matrices and lnverses
	3.3. Determinants as Multilinear Functionals
	3.4. An Alternative Method for Evaluating a Determinant
	3.5. An Introduction to Alternating Multilinear Forms
	3.6. Determinants Discovered Anew
4. Inner-Product Spaces
	4.1. Inner Products in V_3
	4.2. General Euclidean Spaces
	4.3. The Gram-Schmidt Process of Orthogonalization
	4.4. Orthogonal Complements
	4.5. Orthogonal Transformations
	4.6. Linear Functionals and Adjoints
	4.7. Inner Products and Positive Operators
	4.8. Simple Applications of the Distance Function
5. Bilinear and Quadratic Forms
5.1. Bilinear Functions and Forms
5.2. Quadratic Forms
5.3. Diagonal Quadratic Forms Under Congruence
5.4. Invariants of a Symmetric Matrix Under Congruence
5.5. Eigenvalues and Eigenvectors
5.6. Orthogonal Reduction of Quadratic Forms
6. Similarity and Normal Operators
6.1. The Cayley-Hamilton Theorem
6.2. Similarity and Diagonal Matrices
6.3. Complex Vector Spaces
6.4. The Spectral Theorem
6.5. Invariant Subspaces and Primary Decomposition
6.6. Nilpotent Operators and T-cyclic Subspaces
6.7. The Jordan Canonical Form
Appendix: The Real Case: A Variant of the Jordan Canonical Form
Glossary of Symbols
Answers
Index