Applied Linear Algebra 1st edition

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• Chapter 1: Solving Linear Systems and the Terminology of Vectors and Matrices
  • 1.1: Solving Linear Systems, Gaussian elimination, back substitution
  • 1.2: Column vectors, addition, scalar multiplication, the two geometric interpretations
  • 1.3: Matrices, addition, scalar and matrix multiplication, matrix form of a linear system
  • 1.4: Elementary matrices, permutation matrices, LU factorization
  • 1.5: Inverses, Gauss-Jordan elimination, transposes, symmetric matrices
  
  
• Chapter 2: Vector Spaces, Linear Transformations, and Their Applications
  • 2.1: Vector spaces, subspaces, column space and null space of a matrix
  • 2.2: Finding the column space and null space, Echelon form, general factorization, pivot and free variables, superposition, rank, efficient solution methods
  • 2.3: Linear independence, basis, dimension, coordinates relative to a basis
  • 2.4: The four fundamental subspaces
  • 2.5: Applications to networks
  • 2.6: Linear transformations
  • 2.7: Coordinates of vectors and linear transformations relative to bases
  
  
• Chapter 3: Orthogonality and Projections
  • 3.1: Orthogonal vectors, subspaces, and orthogonal complements
  • 3.2: Projection onto a vector
  • 3.3: Projections onto subspaces and least squares approximation
  • 3.4: Orthogonal bases, orthogonal matrices, Gram-Schmidt
  • 3.5: Function spaces, Fourier series, and orthogonal polynomials
  
  
• Chapter 4: Determinants and Their Applications
  • 4.1: Definition and properties of the determinant
  • 4.2: Formulas for the determinant, cofactor expansions
  • 4.3: Applications of determinants
  
  
• Chapter 5: Diagonalization and Eigenvalues and Eigenvectors
  • 5.1: Eigenvalues, eigenvectors, computation examples
  • 5.2: Diagonalization of a matrix
  • 5.3: Complex numbers and complex eigenvalue problems
  • 5.4: The Google page rank algorithm
  • 5.5: Differential equations and matrix exponentials
  • 5.6: Complex matrices, the spectral theorem, and similarity
  • 5.7: Coordinate change under a change of basis
  
  
• Chapter 6: Quadratic Forms and the Singular Value Decomposition
  • 6.1: Minima, maxima, saddle points, definite and semi-definite quadratic forms
  • 6.2: Tests for positive definiteness
  • 6.3: Singular value factorization and decomposition