Linear Algebra

Index

• 1. Foundations and Prerequisites
  • 1.1 Sets, Functions, and Notation
  • 1.2 Number Systems and Fields
  • 1.3 Proof Techniques in Linear Algebra
• 2. Vectors in Euclidean Space
  • 2.1 Vector Fundamentals
  • 2.2 Vector Operations
  • 2.3 Dot Product and Geometry
  • 2.4 Cross Product and Spatial Geometry
  • 2.5 Lines and Planes
• 3. Systems of Linear Equations
  • 3.1 Linear Systems and Solutions
  • 3.2 Matrix Representation
  • 3.3 Gaussian Elimination
  • 3.4 Structure of Solution Sets
  • 3.5 Applications of Linear Systems
• 4. Matrix Algebra
  • 4.1 Matrices and Basic Operations
  • 4.2 Transpose and Special Matrices
  • 4.3 Matrix Inverses
  • 4.4 Elementary Matrices and Factorizations
  • 4.5 Block Matrices
• 5. Determinants
  • 5.1 Definition and Computation
  • 5.2 Properties of Determinants
  • 5.3 Applications of Determinants
• 6. Vector Spaces
  • 6.1 Abstract Vector Spaces
  • 6.2 Subspaces
  • 6.3 Linear Independence
  • 6.4 Basis and Dimension
  • 6.5 Coordinates and Change of Basis
  • 6.6 Fundamental Subspaces of a Matrix
• 7. Linear Transformations
  • 7.1 Definition and Examples
  • 7.2 Matrix Representations
  • 7.3 Kernel, Image, and Structure
  • 7.4 Spaces of Linear Maps and Duality
  • 7.5 Quotient Spaces
• 8. Eigenvalues and Eigenvectors
  • 8.1 Eigenvalue Fundamentals
  • 8.2 Diagonalization
  • 8.3 Advanced Eigenstructure
  • 8.4 Applications of Eigenvalues
• 9. Inner Product Spaces and Orthogonality
  • 9.1 Inner Products
  • 9.2 Orthogonal Sets and Bases
  • 9.3 Gram-Schmidt and QR
  • 9.4 Orthogonal and Unitary Matrices
  • 9.5 Least Squares Approximation
• 10. Symmetric Matrices and Quadratic Forms
  • 10.1 Spectral Theory
  • 10.2 Quadratic Forms
  • 10.3 Definiteness and Optimization
• 11. Singular Value Decomposition and Matrix Analysis
  • 11.1 Singular Value Decomposition
  • 11.2 Applications of SVD
  • 11.3 Matrix Norms and Conditioning
• 12. Numerical and Computational Linear Algebra
  • 12.1 Direct Methods
  • 12.2 Iterative Methods for Linear Systems
  • 12.3 Eigenvalue Algorithms
• 13. Advanced Topics and Applications
  • 13.1 Multilinear Algebra
  • 13.2 Positive Matrices and Graph Theory
  • 13.3 Linear Algebra in Applied Domains
  • 13.4 Connections to Further Mathematics

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