CHAPTER I

• Vector Spaces
• 1 Definitions
• 2 Bases
• 3 Dimension of a Vector Space
• 4 Sums and Direct Sums

CHAPTER II

• Matrices
• 1 The Space of Matrices
• 2 Linear Equations
• 3 Multiplication of Matrices

CHAPTER III

• Linear Mappings
• 1 Mappings
• 2 Linear Mappings
• 3 The Kernel and Image of a Linear Map
• 4 Composition and Inverse of Linear Mappings
• 5 Geometric Applications

CHAPTER IV

• Linear Maps and Matrices
• 1 The Linear Map Associated with a Matrix
• 2 The Matrix Associated with a Linear Map
• 3 Bases, Matrices, and Linear Maps

CHAPTER V

• Scalar Products and Orthogonality
• 1 Scalar Products
• 2 Orthogonal Bases, Positive Definite Case
• 3 Application to Linear Equations, the Rank
• 4 Bilinear Maps and Matrices
• 5 General Orthogonal Bases
• 6 The Dual Space and Scalar Products
• 7 Quadratic Forms
• 8 Sylvester's Theorem

CHAPTER VI

• Determinants
• 1 Determinants of Order
• 2 Existence of Determinants
• 3 Additional Properties of Determinants
• 4 Cramer's Rule
• 5 Triangulation of a Matrix by Column Operations
• 6 Permutations
• 7 Expansion Formula and Uniqueness of Determinants
• 8 Inverse of a Matrix
• 9 The Rank of a Matrix and Subdeterminants

CHAPTER VII

• Symmetric, Hermitian, and Unitary Operators
• 1 Symmetric Operators
• 2 Hermitian Operators
• 3 Unitary Operators

CHAPTER VIII

• Eigenvectors and Eigenvalues
• 1 Eigenvectors and Eigenvalues
• 2 The Characteristic Polynomial
• 3 Eigenvalues and Eigenvectors of Symmetric Matrices
• 4 Diagonalization of a Symmetric Linear Map
• 5 The Hermitian Case
• 6 Unitary Operators

CHAPTER IX

• Polynomials and Matrice
• 1 Polynomials
• 2 Polynomials of Matrices and Linear Maps

CHAPTER X

• Triangulation of Matrices and Linear Maps
• 1 Existence of Triangulation
• 2 Theorem of Hamilton-Cayley
• 3 Diagonalization of Unitary Maps

CHAPTER XI Polynomials and Primary Decomposition

• 1 The Euclidean Algorithm
• 2 Greatest Common Divisor
• 3 Unique Factorization
• 4 Application to the Decomposition of a Vector Space
• 5 Schur's Lemma
• 6 The Jordan Normal Form

CHAPTER XII Convex Sets

• 1 Definitions
• 2 Separating Hyperplanes
• 3 Extreme Points and Supporting Hyperplanes
• 4 The Krein-Milman Theorem