Linear Algebra: A Modern Introduction 5th Edition By David Poole
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Linear Algebra: A Modern Introduction 5th Edition By David Poole

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About this ebook
Linear Algebra: A Modern Introduction (5th Edition) by David Poole covers eight core chapters designed to transition students from computational matrix calculations to abstract, theoretical mathematics. [1, 2]
The structural blueprint of the topics covered in this edition includes:
1. Vectors
  • The Geometry and Algebra of Vectors: Vector operations, geometric interpretations, and linear combinations.
  • Length and Angle: The dot product, norm, distance formula, and Cauchy-Schwarz inequality.
  • Lines and Planes: Vector and parametric equations of paths and surfaces in \(\mathbb{R}^{n}\).
  • Applications: Code vectors, force vectors, and modern optimization models. [1, 2, 3, 4, 5]
2. Systems of Linear Equations
  • Introduction to Systems: Identifying unique, infinite, or no-solution systems geometrically.
  • Direct Methods for Solving: Gaussian elimination, Gauss-Jordan elimination, and row echelon forms.
  • Spanning Sets and Linear Independence: Conceptual foundational blocks for abstract spaces.
  • Applications: Allocation of resources, network analysis, and chemical balancing. [1, 2, 3, 4]
3. Matrices
  • Matrix Operations: Matrix addition, scalar multiplication, and matrix-matrix multiplication.
  • Matrix Algebra: Properties of matrices, the transpose, and symmetric matrices.
  • The Matrix Inverse: Computing inverses and understanding invertible matrix properties.
  • The LU Factorization: Decomposing matrices for efficient computer architectures. [1, 2, 3, 4, 5]
4. Eigenvalues and Eigenvectors
  • Introduction to Eigenvalues/Eigenvectors: Determinants, characteristic polynomials, and eigenspaces.
  • Similarity and Diagonalization: Transforming matrices into simpler diagonal structures.
  • Iterative Methods: Power method for calculating dominant eigenvalues.
  • Applications: Markov chains, population growth models (Leslie matrices), and Google's PageRank algorithm. [1, 2, 3, 4, 5]
5. Orthogonality
  • Orthogonality in \(\mathbb{R}^{n}\): Orthogonal complements, orthogonal bases, and projections.
  • The Gram-Schmidt Process: Constructing orthogonal bases and the QR Factorization.
  • Orthogonal Matrices: Properties and uses in geometric rotations.
  • Applications: Dual codes and basic signal compression. [1, 2, 3]
6. Vector Spaces
  • Vector Spaces and Subspaces: Axioms, definitions, and verification of general spaces.
  • Linear Independence, Basis, and Dimension: Coordinates, coordinate vectors, and the dimension theorem.
  • Null Space and Column Space: Finding bases for fundamental matrix spaces.
  • Linear Transformations: Kernel, range, and building matrices of transformations. [1, 2, 3, 4, 5]
7. Distance and Approximation
  • Inner Product Spaces: Generalizing dot products to functions and matrices.
  • Least Squares Approximation: Finding the best fit for overdetermined systems.
  • The Singular Value Decomposition (SVD): Decomposing any matrix into fundamental building blocks for modern data science. [1, 2, 3]
8. Codes
  • Error-Detecting and Error-Correcting Codes: Linear codes, parity-check matrices, and Hamming codes. [1, 2, 3]

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Topics

Applications: Allocation of resources network analysis and chemical balancing Applications: Code vectors force vectors and modern optimization models Applications: Dual codes and basic signal compression. Applications: Markov chains population growth models (Leslie matrices) and Google's PageRank algorithm Error-Detecting and Error-Correcting Codes: Linear codes parity-check matrices and Hamming codes Inner Product Spaces: Generalizing dot products to functions and matrices Length and Angle: The dot product norm Linear Transformations: Kernel range and building matrices of transformations Matrix Algebra: Properties of matrices the transpose and symmetric matrices Matrix Operations: Matrix addition scalar multiplication and matrix-matrix multiplication Null Space and Column Space: Finding bases for fundamental matrix spaces Spanning Sets and Linear Independence: Conceptual foundational blocks for abstract spaces The Geometry and Algebra of Vectors The LU Factorization: Decomposing matrices for efficient computer architectures The Singular Value Decomposition (SVD): unique infinite or no-solution systems geometrically. Vector and parametric equations of paths and surfaces in \(\mathbb{R}^{n}\) • Direct Methods for Solving: Gaussian elimination Gauss-Jordan elimination and row echelon forms. • Introduction to Eigenvalues/Eigenvectors: Determinants characteristic polynomials and eigenspaces. • Orthogonality in \(\mathbb{R}^{n}\): Orthogonal complements orthogonal bases and projections. • The Gram-Schmidt Process: Constructing orthogonal bases and the QR Factorization. • Vector Spaces and Subspaces: Axioms definitions and verification of general spaces.
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