Linear Algebra With Applications - Emory University

1 With Open TextsLINEAR Algebra with ApplicationsOpen EditionBASE TEXTBOOKVERSION 2019 REVISION AADAPTABLE | ACCESSIBLE | AFFORDABLEby W. Keith NicholsonCreative Commons License (CC BY-NC-SA)advancing learningChampions of Access to KnowledgeOPEN TEXTONLINEASSESSMENTAll digital forms of access to our high-qualityopen texts are entirely FREE! All content isreviewed for excellence and is wholly adapt-able; custom editions are produced by Lyryxfor those adopting Lyryx assessment. Accessto the original source files is also open to any-one!

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4 This text, including the art and illustrations, are availableunder the Creative Commons license (CC BY-NC-SA), allowinganyone to reuse, revise, remix andredistribute the view a copy of this license, learningLinear Algebra with ApplicationsOpen EditionBase Text Revision HistoryCurrent Revision: Version 2019 Revision A2019 A New Section on Singular Value Decomposition ( ) is included. New Please note that this will impact the numbering of subsequentexamples and theorems in the relevant sections.

5 Renamed asMatrix-Vector Multiplication. Minor revisions made throughout, including fixing typos, adding exercises, expanding explanations,and other small B Images have been converted to LaTeX throughout. Text has been converted to LaTeX with minor fixes throughout. Page numbers will differ from 2018 Arevision. Full index has been A Text has been released with a Creative Commons Systems of Linear Solutions and Elementary Operations.. Gaussian Elimination.. Homogeneous Equations.

6 An Application to Network Flow.. An Application to Electrical Networks.. An Application to Chemical Reactions.. 32 Supplementary Exercises for Chapter 1.. 332 Matrix Matrix Addition, Scalar Multiplication, and Transposition.. Matrix-Vector Multiplication.. Matrix Multiplication.. Matrix Inverses.. Elementary Matrices.. Linear Transformations.. LU-Factorization.. An Application to Input-Output Economic Models.. An Application to Markov Chains.. 134 Supplementary Exercises for Chapter 2.

7 1433 Determinants and The Cofactor Expansion.. Determinants and Matrix Inverses.. Diagonalization and Eigenvalues.. An Application to Linear Recurrences.. An Application to Systems of Differential Equations.. Proof of the Cofactor Expansion Theorem.. 204 Supplementary Exercises for Chapter 3.. 2084 Vector Vectors and Lines.. Projections and Planes.. More on the Cross Product.. Linear Operators onR3.. An Application to Computer Graphics.. 258 Supplementary Exercises for Chapter 4.

8 2615 Vector Subspaces and Spanning.. Independence and Dimension.. Orthogonality.. Rank of a Matrix.. Similarity and Diagonalization.. Best Approximation and Least Squares.. An Application to Correlation and Variance.. 322 Supplementary Exercises for Chapter 5.. 3276 Vector Examples and Basic Properties.. Subspaces and Spanning Sets.. Linear Independence and Dimension.. Finite Dimensional Spaces.. An Application to Polynomials.. An Application to Differential Equations.

9 368 Supplementary Exercises for Chapter 6.. 3737 Linear Examples and Elementary Properties.. Kernel and Image of a Linear Transformation.. Isomorphisms and Composition.. A Theorem about Differential Equations.. More on Linear Recurrences.. 4058 Orthogonal Complements and Projections.. Orthogonal Diagonalization.. Positive Definite Matrices.. QR-Factorization.. Computing Eigenvalues.. The Singular Value Decomposition.. Singular Value Decompositions.. Fundamental Subspaces.

10 The Polar Decomposition of a Real Square Matrix.. The Pseudoinverse of a Matrix.. Complex Matrices.. An Application to Linear Codes over Finite Fields.. An Application to Quadratic Forms.. An Application to Constrained Optimization.. An Application to Statistical Principal Component Analysis.. 5009 Change of The Matrix of a Linear Transformation.. Operators and Similarity.. Invariant Subspaces and Direct Sums.. 52210 Inner Product Inner Products and Norms.. Orthogonal Sets of Vectors.