Linear Algebra » Part 5: Euclidean Spaces » Bilinear

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General Public Licence (GPL)

Email: Prof. Vladimir Dobrushkin
Linear Algebra with Mathematica

Introduction to Linear Algebra

  • Linear Algebra Software
  • Notation

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
  • 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
  • Coordinate systems
  • Change of basis
  • Linear transformations
  • Isomorphisms
  • Dual spaces
  • Dual transformations
  • Subspaces
  • Intersections
  • Direct Sums
  • 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
  • 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
  • Sylvester Formula
  • Cholesky decomposition
  • Schur decomposition
  • Jordan decomposition
  • Positive Matrices
  • Roots
  • Polar Factorization
  • Spectral Decomposition
  • Singula 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
  • Differential forms

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

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

  • ABCDEFGHI
  • JKLMNOPQR
  • STUVWXYZ

Reference

  • Free Materials
  • Books
  • GFDL License
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