Introduction to Linear Algebra
Systems of Linear Equations
- Introduction
- Linear systems
- Vectors
- Linear combinations
- Matrices
- Planes in ℝ³
- Row operations
- Gaussian elimination
- Reduced Row-Echelon Form
- Equation A x = b
- Sensitivity of solutions
- Linear independence
- Plane transformations
- Space transformations
- Linear transformations
- Affine maps
- Exercises
- Answers
Matrix Algebra
- Introduction
- Manipulation of matrices
- Matrix transformations
- Block matrices
- Determinants
- Cofactors
- Cramer's rule
- Partitioned matrices
- Elementary matrices
- Inverse matrices
- Elimination: A = LU
- PLU factorization
- Reflection
- Givens rotation
- Special matrices
- Exercises
- Answers
Vector Spaces
- Introduction
- Motivation
- Vector spaces
- Bases
- Dimension
- Linear transformations
- Compositions
- Change of basis
- Isomorphisms
- Dual spaces
- Dual transformations
- Subspaces
- Intersections
- Direct sums
- Quotient spaces
- Vector products
- Matrix spaces
- Row space
- Range or Column space
- Rank
- Null Spaces or Kernels
- Dimension Theorems
- Four Subspaces
- Solving A x = b
- Exercises
- Answers
Eigenvalues, Eigenvectors
- Introduction
- Characteristic polynomials
- Algebraic and geometric multiplicities
- Minimal polynomials
- Eigenspaces
- Where are eigenvalues?
- Eigenvalues of AB and BA
- Generalized eigenvectors
Euclidean Spaces
- Introduction
- Dot product
- Bilinear transformations
- Inner product
- Norm and distance
- Matrix norms
- Dual norms
- Dual transformations
- Examples of transformations
- Orthogonality
- Gram--Schmidt Process
- Orthogonal sets
- Self-adjoint Matrices
- Unitary matrices
- Projection operators
- QR-decomposition
- Least Square Approximation
- Quadratic forms
- Exercises
- Answers
Matrix Decompositions
- Introduction
- LU-decomposition
- QR-decomposition
- Cholesky decomposition
- Schur decomposition
- Jordan decomposition
- Positive matrices
- Roots
- Polar factorization
- Spectral decomposition
- Singular values
- SVD <
- Pseudoinverse
- Exercises
- Answers
Applications
- GPS Problem
- Graph Theory
- Error Correcting Codes
- Electric Circuits
- Markov Chains
- Cryptography
- Wave-length Transfer Matrix
- Computer Graphics
- Linear Programming
- Hill's Determinant
- Fibonacci Matrices
- Discrete Fourier Transform
- Fast Fourier Transform
- Curve fitting
- TNB frames
- Calculus
Functions of Matrices
- Introduction
- Similar matrices
- Diagonalization
- Sylvester Formula
- The Resolvent method
- Polynomial Interpolation
- Positive Matrices
- Roots <
- Pseudoinverse
- Exercises
- Answers
Miscellany
- Vector Representations
- Matrix Representations
- Change of Basis
- Orthonormal Diagonalization
- Generalized Inverse
- Differential forms
Preliminaries
- Complex Number Operations
- Sets
- Polynomials
- Polynomials and Matrices
- Computer solves Systems of Linear Equations
- Location of Eigenvalues
- Power Method
- Iterative Method
- Similarity and Diagonalization
Glossary
Reference
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Calculus