Linear Algebra

Chapter 1: Introduction to Vector Spaces

Abstraction and properties of scalars, vectors, and vector spaces

• 1.1 Fields and Subfields
• 1.2 Vector Spaces and Subspaces
• 1.3 Span and Linear Independence
• 1.4 Basis and Dimension

Chapter 2: Linear Maps and Matrices

Theory of linear maps, matrix representations, change of basis, and duality

• 2.1 Introduction to Linear Maps
• 2.2 Introduction to Matrices
• 2.3 Invertibility and Isomorphisms
• 2.4 Product and Quotient Spaces

Chapter 3: Linear Systems

Theory of linear systems, Gaussian elimination, related matrix factorizations, and applications

• 3.1 Linear Systems and Matrix Equations
• 3.2 Echelon Forms and Row Reduction
• 3.3 Computation of Matrix Subspaces

Chapter 4: Determinants

Theory of multilinear maps, determinants, properties, and applications

• 4.1 Determinants and Permutations
• 4.2 Properties of Determinants
• 4.3 Cofactor Expansion and Cramer's Rule

Chapter 5: Eigenvalues and Eigenvectors

Theory of eigenvalues, eigenvectors, polynomials, and diagonalization

• 5.1 Invariant Subspaces
• 5.2 Polynomials

Chapter 6: Inner Product Spaces and Orthogonality

Theory of inner product spaces and orthogonality

• 6.1 Inner Products and Norms
• 6.2 Orthogonality and Projections

Chapter 7: Spectral Theory

Theory of spectral decomposition, singular values, and quadratic forms

Chapter 8: Duality and Tensors

Theory of multilinear algebra, duality, and tensors

• 8.1 Dual Spaces and Dual Maps

Chapter 9: Jordan Canonical Form

Theory of generalized eigenspaces and the Jordan canonical form

Chapter 10: Applications of Linear Algebra

Applications of linear algebra to various fields

Chapter 11: Appendix

Solutions to exercises and additional material

• 11.1 Solutions