This volume develops vector spaces, linear maps, matrices, and multilinear structures.
This volume develops vector spaces, linear maps, matrices, and multilinear structures. It serves as a core toolkit for nearly all areas of mathematics, physics, and computation.
Part I. Vector Spaces
Chapter 1. Vector Spaces
1.1 Definitions and examples 1.2 Subspaces 1.3 Linear combinations 1.4 Span and generation 1.5 Linear independence
Chapter 2. Bases and Dimension
2.1 Bases 2.2 Existence of bases 2.3 Dimension 2.4 Coordinate representations 2.5 Change of basis
Chapter 3. Linear Maps
3.1 Definitions 3.2 Kernel and image 3.3 Rank and nullity 3.4 Composition of maps 3.5 Isomorphisms
Part II. Matrix Theory
Chapter 4. Matrices
4.1 Matrix definitions 4.2 Matrix operations 4.3 Matrix multiplication 4.4 Identity and inverse matrices 4.5 Block matrices
Chapter 5. Systems of Linear Equations
5.1 Gaussian elimination 5.2 Row echelon form 5.3 Solutions and consistency 5.4 Rank conditions 5.5 Applications
Chapter 6. Determinants
6.1 Definition and properties 6.2 Cofactor expansion 6.3 Determinant and invertibility 6.4 Geometric interpretation 6.5 Computation
Part III. Eigenvalues and Diagonalization
Chapter 7. Eigenvalues and Eigenvectors
7.1 Definitions 7.2 Characteristic polynomial 7.3 Algebraic and geometric multiplicity 7.4 Computation methods 7.5 Examples
Chapter 8. Diagonalization
8.1 Diagonalizable matrices 8.2 Similarity transformations 8.3 Jordan normal form (overview) 8.4 Applications 8.5 Limitations
Chapter 9. Spectral Theory
9.1 Spectral decomposition 9.2 Normal matrices 9.3 Hermitian and unitary matrices 9.4 Applications 9.5 Extensions
Part IV. Inner Product Spaces
Chapter 10. Inner Products
10.1 Definitions 10.2 Norms and distances 10.3 Orthogonality 10.4 Projections 10.5 Examples
Chapter 11. Orthogonalization
11.1 Gram–Schmidt process 11.2 Orthonormal bases 11.3 QR decomposition 11.4 Applications 11.5 Numerical considerations
Chapter 12. Quadratic Forms
12.1 Definitions 12.2 Diagonalization 12.3 Positive definiteness 12.4 Sylvester’s law 12.5 Applications
Part V. Multilinear Algebra
Chapter 13. Tensor Products
13.1 Definition 13.2 Universal property 13.3 Tensor spaces 13.4 Applications 13.5 Examples
Chapter 14. Exterior Algebra
14.1 Alternating forms 14.2 Wedge product 14.3 Determinants via exterior algebra 14.4 Applications in geometry 14.5 Examples
Chapter 15. Bilinear and Multilinear Forms
15.1 Bilinear maps 15.2 Symmetric and alternating forms 15.3 Dual spaces 15.4 Tensor contractions 15.5 Applications
Part VI. Advanced Matrix Theory
Chapter 16. Matrix Decompositions
16.1 LU decomposition 16.2 QR decomposition 16.3 Singular value decomposition 16.4 Eigen decomposition 16.5 Applications
Chapter 17. Matrix Functions
17.1 Polynomial functions of matrices 17.2 Exponential of a matrix 17.3 Functional calculus (overview) 17.4 Applications 17.5 Computation
Chapter 18. Structured Matrices
18.1 Sparse matrices 18.2 Toeplitz and circulant matrices 18.3 Block structures 18.4 Low-rank approximations 18.5 Applications
Part VII. Computational Linear Algebra
Chapter 19. Numerical Methods
19.1 Floating-point arithmetic 19.2 Stability and conditioning 19.3 Iterative methods 19.4 Error analysis 19.5 Applications
Chapter 20. Large-Scale Computation
20.1 Sparse systems 20.2 Krylov subspace methods 20.3 Parallel computation 20.4 Memory considerations 20.5 Applications
Chapter 21. Applications
21.1 Machine learning 21.2 Data analysis 21.3 Signal processing 21.4 Optimization 21.5 Scientific computing
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Operator theory connections 22.2 Infinite-dimensional extensions 22.3 Random matrix theory 22.4 Tensor methods 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Numerical stability challenges 23.2 High-dimensional computation 23.3 Tensor complexity 23.4 Spectral problems 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of linear algebra 24.2 Key contributors 24.3 Evolution of matrix methods 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Common identities and formulas B. Standard matrix decompositions C. Proof templates D. Algorithm templates E. Cross-reference to other MSC branches
This volume provides the central computational and structural toolkit of mathematics. It connects algebra, geometry, and computation through linear structure.