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<p>Preface xi</p> <p>Features of the Text xiii</p> <p>Acknowledgments xvii</p> <p>About the Companion Website xviii</p> <p><b>1 Systems of Linear Equations 1</b></p> <p>1.1 The Vector Space of m × n Matrices 1</p> <p>The Space ℝⁿ 4</p> <p>Linear Combinations and Linear Dependence 7</p> <p>What Is a Vector Space? 11</p> <p>Why Prove Anything? 15</p> <p>Exercises 16</p> <p>1.1.1 Computer Projects/Exercises/Exercises 22</p> <p>Exercises 24</p> <p>1.1.2 Applications to Graph Theory I 25</p> <p>Exercises 27</p> <p>1.2 Systems 27</p> <p>Rank: The Maximum Number of Linearly Independent Equations 34</p> <p>Exercises 37</p> <p>1.2.1 Computer Projects/Exercises 39</p> <p>Exercises 39</p> <p>1.2.2 Applications to Circuit Theory 40</p> <p>Exercises 44</p> <p>1.3 Gaussian Elimination 46</p> <p>Spanning in Polynomial Spaces 56</p> <p>Computational Issues: Pivoting 59</p> <p>Exercises 60</p> <p>1.3.1 Using tolerances in MATLAB’s rref and rank 66</p> <p>Using Tolerances in rref and Rank 66</p> <p>Exercises 67</p> <p>1.3.2 Applications to Traffic Flow 68</p> <p>Exercises 70</p> <p>1.4 Column Space and Nullspace 71</p> <p>Subspaces 74</p> <p>Exercises 82</p> <p>1.4.1 Computer Projects/Exercises 89</p> <p>Exercises 90</p> <p>Chapter Summary 91</p> <p><b>2 Linear Independence and Dimension 93</b></p> <p>2.1 The Test for Linear Independence 93</p> <p>Bases for the Column Space 100</p> <p>Testing Functions for Independence 102</p> <p>Exercises 104</p> <p>2.1.1 Computer Projects/Exercises 108</p> <p>Exercises 108</p> <p>2.2 Dimension 109</p> <p>Exercises 118</p> <p>2.2.1 Computer Projects/Exercises 123</p> <p>Exercises 123</p> <p>2.2.2 Applications to Differential Equations 125</p> <p>Exercises 128</p> <p>2.3 Row Space and the Rank-Nullity Theorem 128</p> <p>Bases for the Row Space 130</p> <p>Computational Issues: Computing Rank 138</p> <p>Exercises 140</p> <p>2.3.1 Computer Projects/Exercises 143</p> <p>Exercises 143</p> <p>Chapter Summary 144</p> <p><b>3 Linear Transformations 147</b></p> <p>3.1 The Linearity Properties 147</p> <p>Exercises 155</p> <p>3.1.1 Computer Projects/Exercises 160</p> <p>Exercises 161</p> <p>3.2 Matrix Multiplication (Composition) 162</p> <p>Partitioned Matrices 169</p> <p>Computational Issues: Parallel Computing 171</p> <p>Exercises 171</p> <p>3.2.1 Computer Projects/Exercises 177</p> <p>3-D Computer Graphics 177</p> <p>Exercises 177</p> <p>3.2.2 Applications to Graph Theory II 178</p> <p>Exercises 180</p> <p>3.2.3 Computer Projects/Exercises 180</p> <p>Google’s Page Rank Algorithm 180</p> <p>Exercises 183</p> <p>3.3 Inverses 184</p> <p>Computational Issues: Reduction versus Inverses 190</p> <p>Exercises 192</p> <p>3.3.1 Computer Projects/Exercises 197</p> <p>Ill-Conditioned Systems 197</p> <p>Exercises 197</p> <p>3.3.2 Applications to Economics: The Leontief Open Model 199</p> <p>Exercises 204</p> <p>3.4 The LU Factorization 205</p> <p>Exercises 213</p> <p>3.4.1 Computer Projects/Exercises 216</p> <p>Exercises 216</p> <p>3.5 The Matrix of a Linear Transformation 217</p> <p>Coordinates 217</p> <p>Application to Differential Equations 225</p> <p>Isomorphism 228</p> <p>Invertible Linear Transformations 229</p> <p>Exercises 231</p> <p>3.5.1 Computer Projects/Exercises 236</p> <p>Graphing in Skewed-Coordinates 236</p> <p>Exercises 236</p> <p>3.5.2 Computer Projects/Exercises 237</p> <p>Pricing Long Term Health Care Insurance 237</p> <p>Exercises 242</p> <p>Chapter Summary 242</p> <p><b>4 Determinants 245</b></p> <p>4.1 Definition of the Determinant 245</p> <p>4.1.1 The Rest of the Proofs 252</p> <p>Exercises 256</p> <p>4.1.2 Computer Projects/Exercises 258</p> <p>4.2 Reduction and Determinants 259</p> <p>Exercises 266</p> <p>4.2.1 Volume 268</p> <p>Exercises 271</p> <p>4.3 A Formula for Inverses 271</p> <p>Exercises 275</p> <p>Chapter Summary 276</p> <p><b>5 Eigenvectors and Eigenvalues 279</b></p> <p>5.1 Eigenvectors 279</p> <p>Exercises 288</p> <p>5.1.1 Computer Projects/Exercises 291</p> <p>Exercises 291</p> <p>5.1.2 Application to Markov Chains 291</p> <p>Exercises 294</p> <p>5.2 Diagonalization 295</p> <p>Powers of Matrices 297</p> <p>Exercises 299</p> <p>5.2.1 Application to Systems of Differential Equations 301</p> <p>Exercises 304</p> <p>5.3 Complex Eigenvectors 304</p> <p>Complex Vector Spaces 311</p> <p>Exercises 312</p> <p>5.3.1 Computer Projects/Exercises 314</p> <p>Exercises 314</p> <p>Chapter Summary 314</p> <p><b>6 Orthogonality 317</b></p> <p>6.1 The Scalar Product in ℝⁿ 317</p> <p>Orthogonal/Orthonormal Bases and Coordinates 321</p> <p>Exercises 326</p> <p>6.2 Projections: The Gram–Schmidt Process 328</p> <p>The QR Decomposition 334</p> <p>Uniqueness of the QR Factorization 337</p> <p>Exercises 338</p> <p>6.2.1 Computer Projects/Exercises 341</p> <p>Exercises 342</p> <p>6.3 Fourier Series: Scalar Product Spaces 342</p> <p>Exercises 350</p> <p>6.3.1 Computer Projects/Exercises 353</p> <p>Exercises 354</p> <p>6.4 Orthogonal Matrices 355</p> <p>Householder Matrices 360</p> <p>Exercises 364</p> <p>6.4.1 Computer Projects/Exercises 369</p> <p>Exercises 369</p> <p>6.5 Least Squares 370</p> <p>Exercises 377</p> <p>6.5.1 Computer Projects/Exercises 380</p> <p>Exercises 380</p> <p>6.6 Quadratic Forms: Orthogonal Diagonalization 381</p> <p>The Spectral Theorem 384</p> <p>The Principal Axis Theorem 385</p> <p>Exercises 392</p> <p>6.6.1 Computer Projects/Exercises 394</p> <p>Exercises 395</p> <p>6.7 The Singular Value Decomposition (SVD) 396</p> <p>Application of the SVD to Least-Squares Problems 402</p> <p>Exercises 404</p> <p>Computing the SVD Using Householder Matrices 406</p> <p>Diagonalizing Matrices Using Householder Matrices 408</p> <p>6.8 Hermitian Symmetric and Unitary Matrices 409</p> <p>Exercises 416</p> <p>Chapter Summary 418</p> <p><b>7 Generalized Eigenvectors 421</b></p> <p>7.1 Generalized Eigenvectors 421</p> <p>Exercises 429</p> <p>7.2 Chain Bases 431</p> <p>Jordan Form 438</p> <p>Exercises 443</p> <p>The Cayley–Hamilton Theorem 444</p> <p>Chapter Summary 445</p> <p><b>8 Numerical Techniques 447</b></p> <p>8.1 Condition Number 447</p> <p>Condition Number 449</p> <p>Least Squares 452</p> <p>Exercises 453</p> <p>8.2 Computing Eigenvalues 454</p> <p>Iteration 454</p> <p>The QR Method 458</p> <p>Exercises 464</p> <p>Chapter Summary 465</p> <p>Answers and Hints 467</p> <p>Index 491</p>

<p><b>RICHARD C. PENNEY, P<small>H</small>D</b> is Emeritus Professor in the Department of Mathematics and former Director of the Mathematics/Statistics Actuarial Science Program at Purdue University. He has authored numerous journal articles, received several major teaching awards, and is an active researcher. He received his graduate education at MIT.

<p><b>Praise for the</b> <b><i>Third Edition</i></b> <p>"This volume is ground-breaking in terms of mathematical texts in that it does not teach from a detached perspective, but instead, looks to show students that competent mathematicians bring an intuitive understanding to the subject rather than just a master of applications."</br> —<b><i>Electric Review</i></b> <p><b>Learn foundational and advanced topics in linear algebra with this concise and approachable resource</b> <p>A comprehensive introduction, <i>Linear Algebra: Ideas and Applications, Fifth Edition</i> provides a discussion of the theory and applications of linear algebra that blends abstract and computational concepts. With a focus on the development of mathematical intuition, the book emphasizes the need to understand both the applications of a particular technique and the mathematical ideas underlying the technique. <p>The book introduces each new concept in the context of explicit numerical examples, which allows the abstract concepts to grow organically out of the necessity to solve specific problems. The intuitive discussions are consistently followed by rigorous statements of results and proofs. <i>Linear Algebra: Ideas and Applications, Fifth Edition</i> also features: <ul> <li>A new application section on Google's Page Rank Algorithm.</li> <li>A new application section on pricing long term health insurance at a Continuing Care Retirement Community (CCRC).</li> <li>Many other illuminating applications of linear algebra with self-study questions for additional study.</li> <li>End-of-chapter summaries and sections with true-false questions to aid readers with further comprehension of the presented material</li> <li>Numerous computer exercises throughout using MATLAB^® code</li> </ul> <p><i>Linear Algebra: Ideas and Applications, Fifth Edition</i> is an excellent undergraduate-level textbook for one or two semester undergraduate courses in mathematics, science, computer science, and engineering. With an emphasis on intuition development, the book is also an ideal self-study reference.

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