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Linear Algebra
Introduction
Main content
1. Vectors, Lines and Planes
1.1. Vectors
1.2. Dot Product
1.3. Cross Product
1.4. Lines and Planes
2. Systems of Linear Equations, Vector Equations and Matrix Equations
2.1. Systems of Linear Equations
2.2. Linear Combinations
2.3. The Solution Set of a System of Linear Equations
2.4. The Matrix-Vector Product
\(A\vect{x}\)
2.5. Linear Independence
3. Matrix Operations
3.1. Linear Transformations
3.2. Matrix Operations
3.3. Some Important Classes of Linear Transformations
3.4. The Inverse of a Matrix
3.5. Injectivity, Surjectivity, and Bijectivity
3.6. Matrix Factorisation
4. Subspaces
4.1. Subspaces of
\(\R^n\)
4.2. Basis and Dimension
4.3. Change of Basis
5. Determinants
5.1. Determinants as Areas or Volumes
5.2. Determinants via Cofactor Expansion
5.3. Determinants via Row Reduction
5.4. Miscellaneous Applications of Determinants
6. Eigenvalues and Eigenvectors
6.1. Definitions and Examples
6.2. The Characteristic Polynomial
6.3. Diagonalizability
6.4. Complex Eigenvalues (and Eigenvectors)
7. Orthogonality
7.1. Orthogonal Complements
7.2. Orthogonal and Orthonormal bases
7.3. The Gram-Schmidt Process
7.4. Least Squares Solutions
8. Symmetric Matrices
8.1. Symmetric Matrices
8.2. Quadratic Forms
8.3. Singular Value Decomposition (SVD)
9. Dynamical Systems
9.1. Discrete Dynamical Systems
9.2. Markov Chains
9.3. The Power Method
9.4. Continuous Dynamical Systems
Appendices
10. Complex numbers
11. Proof of the existence of SVD
12. The inverse matrix theorem
Colophon
Acknowledgements
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Linear Algebra
Linear Algebra
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