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Solutions Manual for Linear Algebra A Modern Introduction 4th Edition by David Poole 2024 . All Chapters A+ Contents 1 Vectors 3 1.1 The Geometry and Algebra of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Length and Angle: The Dot Product . . . . . . . . . . . . ...

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Solutions Manual for Linear Algebra A Modern Introduction 4th Edition by David Poole
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Complete Solutions Manual


Linear Algebra
A Modern Introduction
FOURTH EDITION



David Poole
Trent University




Prepared by

Roger Lipsett




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Full all chapters instant download please go to Solutions Manual, Test Bank site: downloadlink.org

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1 2 3 4 5 6 7 17 16 15 14 13

,Contents

1 Vectors 3
1.1 The Geometry and Algebra of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Length and Angle: The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Exploration: Vectors and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3 Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Exploration: The Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2 Systems of Linear Equations 53
2.1 Introduction to Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 Direct Methods for Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Exploration: Lies My Computer Told Me . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Exploration: Partial Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Exploration: An Introduction to the Analysis of Algorithms . . . . . . . . . . . . . . . . . . . . . . 77
2.3 Spanning Sets and Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.5 Iterative Methods for Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3 Matrices 129
3.1 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.2 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.3 The Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.4 The LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
3.5 Subspaces, Basis, Dimension, and Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.6 Introduction to Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
3.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

4 Eigenvalues and Eigenvectors 235
4.1 Introduction to Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
4.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Exploration: Geometric Applications of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . 263
4.3 Eigenvalues and Eigenvectors of n × n Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 270
4.4 Similarity and Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
4.5 Iterative Methods for Computing Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
4.6 Applications and the Perron-Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 326
Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

1

, 2 CONTENTS

5 Orthogonality 371
5.1 Orthogonality in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
5.2 Orthogonal Complements and Orthogonal Projections . . . . . . . . . . . . . . . . . . . . . . 379
5.3 The Gram-Schmidt Process and the QR Factorization . . . . . . . . . . . . . . . . . . . . . . 388
Exploration: The Modified QR Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Exploration: Approximating Eigenvalues with the QR Algorithm . . . . . . . . . . . . . . . . . . . 402
5.4 Orthogonal Diagonalization of Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . 405
5.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

6 Vector Spaces 451
6.1 Vector Spaces and Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
6.2 Linear Independence, Basis, and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
Exploration: Magic Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
6.3 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
6.4 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
6.5 The Kernel and Range of a Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . 498
6.6 The Matrix of a Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
Exploration: Tiles, Lattices, and the Crystallographic Restriction . . . . . . . . . . . . . . . . . . . 525
6.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

7 Distance and Approximation 537
7.1 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
Exploration: Vectors and Matrices with Complex Entries . . . . . . . . . . . . . . . . . . . . . . . 546
Exploration: Geometric Inequalities and Optimization Problems . . . . . . . . . . . . . . . . . . . 553
7.2 Norms and Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
7.3 Least Squares Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
7.4 The Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
7.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

8 Codes 633
8.1 Code Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
8.2 Error-Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
8.3 Dual Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
8.4 Linear Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647
8.5 The Minimum Distance of a Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

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