[Table of Content of Linear Algebra in PDF format](https://link.springer.com/content/pdf/bfm:978-1-4757-1949-9/1?pdf=chapter%20toc) by [[Serge Lang]] # Table of Content 1. Vector Spaces 1. Definitions 2. Bases 3. Dimension of a Vector Space 4. Sums and Direct Sums 2. Matrices 1. The Space of Matrices 2. Linear Equations 3. Multiplication of Matrices 3. Linear Mappings 1. Mappings 2. Linear Mappings 3. The Kernel and Image of a Linear Map 4. Composition and Inverse of Linear Mapping 5. Geometric Applications 4. 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 5. 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. Determinants 1. Determinants of Order 2 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 7. Symmetric, Hermitian, and Unitary Operators 1. Symmetric Operators 2. Hermitian Operators 3. Unitary Operators 8. Eigenvectors and Eigenvalues 1. Eigenvectors and Eigenvalues 2. The Characteristic Polynomials 3. Eigenvalues and Eigenvectors of Symmetric Matrices 4. Diagonalization of a Symmetric Linear Map 5. The Hermitian Case 6. Unitary Operators 9. Polynomials and Matrices 1. Polynomials 2. Polynomials of Matrices and Linear Maps 10. Triangulation of Matrices and Linear Maps 1. Existence of Triangulation 2. Theorem of Hamilton-Cayley 3. Diagonalization of Unitary Maps 11. 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 12. Convex Sets 1. Definitions 2. Separating Hyperplanes 3. Extreme Points and Support Hyperplanes 4. The Krein-Milman Theorem Appendix I Complex Numbers Appendix II Iwasawa Decomposition and Others # A Diagram ToC in Canvas form ![[Linear Algebra by Serge Lang.canvas]]