Preface |
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xi | |
Features Of The Text |
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xiii | |
Acknowledgments |
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xvii | |
About The Companion Website |
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xviii | |
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1 Systems Of Linear Equations |
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1 | (92) |
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1.1 The Vector Space of m × n Matrices |
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1 | (26) |
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4 | (3) |
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Linear Combinations and Linear Dependence |
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7 | (4) |
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11 | (4) |
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15 | (1) |
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16 | (6) |
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1.1.1 Computer Projects/Exercises/Exercises |
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22 | (2) |
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24 | (1) |
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1.1.2 Applications to Graph Theory I |
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25 | (2) |
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27 | (1) |
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27 | (19) |
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Rank: The Maximum Number of Linearly Independent Equations |
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34 | (3) |
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37 | (2) |
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1.2.1 Computer Projects/Exercises |
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39 | (1) |
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39 | (1) |
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1.2.2 Applications to Circuit Theory |
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40 | (4) |
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44 | (2) |
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46 | (25) |
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Spanning in Polynomial Spaces |
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56 | (3) |
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Computational Issues: Pivoting |
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59 | (1) |
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60 | (6) |
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1.3.1 Using tolerances in MATLAB's rref and rank |
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66 | (1) |
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Using Tolerances in rref and Rank |
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66 | (1) |
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67 | (1) |
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1.3.2 Applications to Traffic Flow |
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68 | (2) |
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70 | (1) |
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1.4 Column Space and Nullspace |
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71 | (22) |
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74 | (8) |
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82 | (7) |
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1.4.1 Computer Projects/Exercises |
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89 | (1) |
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90 | (1) |
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91 | (2) |
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2 Linear Independence And Dimension |
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93 | (54) |
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2.1 The Test for Linear Independence |
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93 | (16) |
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Bases for the Column Space |
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100 | (2) |
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Testing Functions for Independence |
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102 | (2) |
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104 | (4) |
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2.1.1 Computer Projects/Exercises |
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108 | (1) |
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108 | (1) |
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109 | (19) |
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118 | (5) |
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2.2.1 Computer Projects/Exercises |
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123 | (1) |
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123 | (2) |
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2.2.2 Applications to Differential Equations |
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125 | (3) |
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128 | (1) |
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2.3 Row Space and the Rank-Nullity Theorem |
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128 | (19) |
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130 | (8) |
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Computational Issues: Computing Rank |
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138 | (2) |
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140 | (3) |
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2.3.1 Computer Projects/Exercises |
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143 | (1) |
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143 | (1) |
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144 | (3) |
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147 | (98) |
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3.1 The Linearity Properties |
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147 | (15) |
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155 | (5) |
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3.1.1 Computer Projects/Exercises |
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160 | (1) |
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161 | (1) |
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3.2 Matrix Multiplication (Composition) |
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162 | (22) |
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169 | (2) |
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Computational Issues: Parallel Computing |
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171 | (1) |
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171 | (6) |
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3.2.1 Computer Projects/Exercises |
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177 | (1) |
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177 | (1) |
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177 | (1) |
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3.2.2 Applications to Graph Theory II |
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178 | (2) |
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180 | (1) |
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3.2.3 Computer Projects/Exercises |
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180 | (1) |
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Google's Page Rank Algorithm |
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180 | (3) |
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183 | (1) |
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184 | (21) |
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Computational Issues: Reduction versus Inverses |
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190 | (2) |
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192 | (5) |
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3.3.1 Computer Projects/Exercises |
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197 | (1) |
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197 | (1) |
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197 | (2) |
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3.3.2 Applications to Economics: The Leontief Open Model |
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199 | (5) |
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204 | (1) |
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205 | (12) |
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213 | (3) |
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3.4.1 Computer Projects/Exercises |
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216 | (1) |
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216 | (1) |
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3.5 The Matrix of a Linear Transformation |
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217 | (28) |
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217 | (8) |
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Application to Differential Equations |
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225 | (3) |
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228 | (1) |
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Invertible Linear Transformations |
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229 | (2) |
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231 | (5) |
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3.5.1 Computer Projects/Exercises |
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236 | (1) |
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Graphing in Skewed-Coordinates |
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236 | (1) |
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236 | (1) |
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3.5.2 Computer Projects/Exercises |
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237 | (1) |
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Pricing Long Term Health Care Insurance |
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237 | (5) |
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242 | (1) |
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242 | (3) |
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245 | (34) |
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4.1 Definition of the Determinant |
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245 | (14) |
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4.1.1 The Rest of the Proofs |
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252 | (4) |
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256 | (2) |
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4.1.2 Computer Projects/Exercises |
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258 | (1) |
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4.2 Reduction and Determinants |
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259 | (12) |
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266 | (2) |
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268 | (3) |
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271 | (1) |
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4.3 A Formula for Inverses |
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271 | (8) |
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275 | (1) |
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276 | (3) |
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5 Eigenvectors And Eigenvalues |
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279 | (38) |
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279 | (16) |
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288 | (3) |
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5.1.1 Computer Projects/Exercises |
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291 | (1) |
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291 | (1) |
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5.1.2 Application to Markov Chains |
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291 | (3) |
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294 | (1) |
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295 | (9) |
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297 | (2) |
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299 | (2) |
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5.2.1 Application to Systems of Differential Equations |
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301 | (3) |
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304 | (1) |
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304 | (13) |
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311 | (1) |
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312 | (2) |
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5.3.1 Computer Projects/Exercises |
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314 | (1) |
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314 | (1) |
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314 | (3) |
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317 | (104) |
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6.1 The Scalar Product in Rn |
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317 | (11) |
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Orthogonal/Orthonormal Bases and Coordinates |
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321 | (5) |
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326 | (2) |
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6.2 Projections: The Gram-Schmidt Process |
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328 | (14) |
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334 | (3) |
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Uniqueness of the QR Factorization |
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337 | (1) |
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338 | (3) |
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6.2.1 Computer Projects/Exercises |
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341 | (1) |
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342 | (1) |
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6.3 Fourier Series: Scalar Product Spaces |
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342 | (13) |
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350 | (3) |
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6.3.1 Computer Projects/Exercises |
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353 | (1) |
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354 | (1) |
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355 | (15) |
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360 | (4) |
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364 | (5) |
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6.4.1 Computer Projects/Exercises |
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369 | (1) |
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369 | (1) |
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370 | (11) |
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377 | (3) |
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6.5.1 Computer Projects/Exercises |
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380 | (1) |
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380 | (1) |
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6.6 Quadratic Forms: Orthogonal Diagonalization |
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381 | (15) |
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384 | (1) |
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The Principal Axis Theorem |
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385 | (7) |
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392 | (2) |
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6.6.1 Computer Projects/Exercises |
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394 | (1) |
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395 | (1) |
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6.7 The Singular Value Decomposition (SVD) |
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396 | (13) |
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Application of the SVD to Least-Squares Problems |
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402 | (2) |
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404 | (2) |
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Computing the SVD Using Householder Matrices |
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406 | (2) |
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Diagonalizing Matrices Using Householder Matrices |
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408 | (1) |
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6.8 Hermitian Symmetric and Unitary Matrices |
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409 | (12) |
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416 | (2) |
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418 | (3) |
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7 Generalized Eigenvectors |
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421 | (26) |
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7.1 Generalized Eigenvectors |
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421 | (10) |
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429 | (2) |
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431 | (16) |
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438 | (5) |
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443 | (1) |
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The Cayley-Hamilton Theorem |
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444 | (1) |
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445 | (2) |
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447 | (44) |
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447 | (7) |
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449 | (3) |
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452 | (1) |
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453 | (1) |
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8.2 Computing Eigenvalues |
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454 | (37) |
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454 | (4) |
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458 | (6) |
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464 | (1) |
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465 | (2) |
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467 | (24) |
Index |
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491 | |