Introduction to Linear Algebra
Fundamentals
- 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
- Iterative methods
- Linear independence
- Plane transformations
- Space transformations
- Linear transformations
- Affine maps
- Exercises
- Answers
Computations
- Introduction
- Numbers
- 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
Direct Methods
- Introduction
- Motivation
- Vector Spaces
- Bases
- Dimension
- Coordinate systems
- Change of basis
- Linear transformations
- Compositions
- Isomorphisms
- Dual spaces
- Dual transformations
- Subspaces
- Intersections
- Direct sums
- Quotient spaces
- Vector products
- Cross products
- Matrix spaces
- Row space
- Range or Column space
- Rank
- Null spaces or Kernels
- Dimension Theorems
- Four subspaces
- Solving A x = b
- Exercises
- Answers
Iterative Methods
- Introduction
- Richardson's algorithm
- Jacpbi's algorithm
- Gauss--Seidel method
- SOR method
- SSOR method
- Convergence
- Gradient method
- Conjugate gradient
- Krylov subspaces
Eigenvalues
- Introduction
- Dot product
- Bilinear transformations
- Inner product
- Norm and distance
- Matrix norms
- Dual norms
- Dual transformations
- Orthogonality
- Gram--Schmidt Process
- Orthogonal sets
- Self-adjoint matrices
- Unitary matrices
- Projection operators
- QR-decomposition
- Least Square Approximation
- Quadratic forms
- Exercises
- Answers
Orthogonality
- Introduction
- LU-decomposition
- Sylvester Formula
- Cholesky decomposition
- Schur decomposition
- Jordan decomposition
- Positive Matrices
- Roots
- Polar Factorization
- Spectral Decomposition
- Singular values
- SVD <
- Pseudoinverse
- Exercises
- Answers
Matrix Algebra
- 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
Least Squares
- Introduction
- Similar matrices
- Diagonalization
- Sylvester Formula
- The Resolvent Method
- Polynomial Interpolation
- Positive Matrices
- Roots <
- Pseudoinverse
- Exercises
- Answers
Miscellany
- Circles along curves
- TNB frames
- Tensors I
- Tensors II
- Differential forms
- Calculus
- Vector Representations
- Matrix Representations
- Change of Basis
- Orthonormal Diagonalization
- Generalized Inverse
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
This Book is licensed under Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License
Condition Numbers
Let us consider a linear system of equations that we write in concise form A x = b, where A is a square invertible matrix (this condition gquarantees a unique solution), x is a column vector of unknowns, and b is a given column vector. Now suppose we add a small vector δb to b and consider the pertured system A z/b> = b + δb. This system also has a unique solution z, which we hope is not far away from x/b>. Let δx denote the difference between z and x/b>, so z = x + δx.
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