1 Introduction to Matrices
 1.1 Definition of a Matrix
  1.1.1 Special Matrices
 1.2 Operations on Matrices
  1.2.1 Multiplication of Matrices
  1.2.2 Inverse of a Matrix
 1.3 Some More Special Matrices
  1.3.1 Submatrix of a Matrix
 1.4 Summary
2 System of Linear Equations
 2.1 Introduction
  2.1.1 Elementary Row Operations
 2.2 Main Ideas of Linear Systems
  2.2.1 Elementary Matrices and the Row-Reduced Echelon Form (RREF)
  2.2.2 Rank of a Matrix
  2.2.3 Solution set of a Linear System
 2.3 Square Matrices and Linear Systems DRAFT
  2.3.1 Determinant
  2.3.2 Adjugate (classical Adjoint) of a Matrix
  2.3.3 Cramer’s Rule
 2.4 Miscellaneous Exercises
 2.5 Summary
3 Vector Spaces
 3.1 Vector Spaces: Definition and Examples
  3.1.1 Subspaces
  3.1.2 Linear Span
 3.2 Fundamental Subspaces Associated with a Matrix
 3.3 Linear Independence
  3.3.1 Basic Results on Linear Independence
  3.3.2 Application to Matrices
  3.3.3 Linear Independence and Uniqueness of Linear Combination
 3.4 Basis of a Vector Space
  3.4.1 Main Results associated with Bases
  3.4.2 Constructing a Basis of a Finite Dimensional Vector Space
 3.5 Application to the subspaces of ℂⁿ
 3.6 Ordered Bases
 3.7 Summary
4 Linear Transformations
 4.1 Definitions and Basic Properties
 4.2 Rank-Nullity Theorem
  4.2.1 Algebra of Linear Transformations
 4.3 Matrix of a linear transformation
 4.4 Similarity of Matrices
 4.5 Dual Space*
 4.6 Summary
5 Inner Product Spaces
 5.1 Definition and Basic Properties
  5.1.1 Cauchy Schwartz Inequality
  5.1.2 Angle between two Vectors
  5.1.3 Normed Linear Space
 5.2 Gram-Schmidt Orthonormalization Process
  5.2.1 Application to Fundamental Spaces
  5.2.2 QR Decomposition^*
 5.3 Orthogonal Projections and Applications
  5.3.1 Orthogonal Projections as Self-Adjoint Operators*
 5.4 Orthogonal Operator and Rigid Motion
 5.5 Summary
6 Eigenvalues, Eigenvectors and Diagonalizability
 6.1 Introduction and Definitions
  6.1.1 Spectrum of a Matrix DRAFT
 6.2 Diagonalization
  6.2.1 Schur’s Unitary Triangularization
  6.2.2 Diagonalizability of some Special Matrices
  6.2.3 Cayley Hamilton Theorem
 6.3 Quadratic Forms
  6.3.1 Sylvester’s law of inertia
  6.3.2 Applications in Eculidean Plane and Space
7 Jordan Canonical form
 7.1 Jordan Canonical form theorem
 7.2 Minimal polynomial
 7.3 Applications of Jordan Canonical Form
  7.3.1 Coupled system of linear differential equations
  7.3.2 Commuting matrices
8 Advanced Topics on Diagonalizability and Triangularization^*
 8.1 More on the Spectrum of a Matrix
 8.2 Methods for Tridiagonalization and Diagonalization
  8.2.1 Plane Rotations
  8.2.2 Householder Matrices
  8.2.3 Schur’s Upper Triangularization Revisited
 8.3 Commuting Matrices and Simultaneous Diagonalization
  8.3.1 Diagonalization and Real Orthogonal Matrix
  8.3.2 Convergent and nilpotent matrices
9 Appendix
 9.1 Uniqueness of RREF
 9.2 Permutation/Symmetric Groups
 9.3 Properties of Determinant
 9.4 Dimension of W₁ + W₂
 9.5 When does Norm imply Inner Product
 9.6 Roots of a Polynomials
 9.7 Variational characterizations of Hermitian Matrices
Index