Introduction to Linear Algebra with Applications

Over the last few decades, linear algebra has become more relevant than ever. Applications have increased not only in quantity but also in diversity, with linear systems being used to solve problems in chemistry, engineering, economics, nutrition, urban planning, and more. DeFranza and Gagliardi introduce students to the topic in a clear, engaging, and easy-to-follow manner. Topics are developed fully before moving on to the next through a series of natural connections. The result is a solid introduction to linear algebra for undergraduates' first course. Outstanding features include: Early coverage of vector spaces, providing the abstract theory necessary to understand applications Exercises that range from routine to more challenging, extending the concepts and techniques by asking students to construct complete arguments Numerous examples designed to develop intuition and prepare readers to think conceptually about topics as they are introduced Fact summaries to end each chapter that use nontechnical language to recapitulate details and formulas Not-for-sale instructor resource material available to college and university faculty only; contact publisher directly. Brief Table of Contents 1. Systems of Linear Equations and Matrices 2. Linear Combinations and Linear Independence 3. Vector Spaces 4. Linear Transformations 5. Eigenvalues and Eigenvectors 6. Inner Product Spaces Title Page 4 About the Authors 8 Contents 9 Preface 12 To The Student 19 Applications Index 20 Chapter 1: Systems of Linear Equations and Matrices 22 1.1 Systems of Linear Equations 23 The Elimination Method 25 Exercise Set 1.1 33 1.2 Matrices and Elementary Row Operations 35 Solving Linear Systems with Augmented Matrices 37 Echelon Form of a Matrix 38 Exercise Set 1.2 44 1.3 Matrix Algebra 47 Matrix Multiplication 50 Transpose of a Matrix 56 Exercise Set 1.3 58 1.4 The Inverse of a Square Matrix 60 Exercise Set 1.4 66 1.5 Matrix Equations 69 Exercise Set 1.5 72 1.6 Determinants 75 Properties of Determinants 78 Cramer’s Rule 83 Exercise Set 1.6 86 1.7 Elementary Matrices and LU Factorization 89 Elementary Matrices 90 The Inverse of an ElementaryMatrix 92 LU Factorization 93 Solving a Linear System Using LU Factorization 96 PLU Factorization 97 Exercise Set 1.7 98 1.8 Applications of Systems of Linear Equations 100 Balancing Chemical Equations 100 Network Flow 100 Nutrition 102 Economic Input-Output Models 103 Exercise Set 1.8 105 Review Exercises for Chapter 1 110 Chapter 1: Chapter Test 111 Chapter 2: Linear Combinations and Linear Independence 114 2.1 Vectors in R superscript n 115 Exercise Set 2.1 120 2.2 Linear Combinations 122 Vector Form of a Linear System 127 Matrix Multiplication 128 Exercise Set 2.2 129 2.3 Linear Independence 132 Linear Systems 138 Linear Independence and Determinants 139 Exercise Set 2.3 141 Review Exercises for Chapter 2 144 Chapter 2: Chapter Test 146 Chapter 3: Vector Spaces 148 3.1 Definition of a Vector Space 150 Exercise Set 3.1 158 3.2 Subspaces 161 Span of a Set of Vectors 167 The Null Space and Column Space of aMatrix 173 Exercise Set 3.2 175 3.3 Basis and Dimension 177 Dimension 185 Finding a Basis 187 Exercise Set 3.3 192 3.4 Coordinates and Change of Basis 194 Change of Basis 198 Inverse of a Transition Matrix 202 Exercise Set 3.4 203 3.5 Application: Differential Equations 206 The Exponential Model 207 Second-Order Differential Equations with Constant Coefficients 207 Fundamental Sets of Solutions 209 Exercise Set 3.5 214 Review Exercises for Chapter 3 215 Chapter 3: Chapter Test 216 Chapter 4: Linear Transformations 220 4.1 Linear Transformations 221 Operations with Linear Transformations 230 Exercise Set 4.1 232 4.2 The Null Space and Range 235 Matrices 242 Linear Systems 243 Exercise Set 4.2 244 4.3 Isomorphisms 247 Exercise Set 4.3 254 4.4 Matrix Representation of a Linear Transformation 256 Exercise Set 4.4 266 4.5 Similarity 270 Exercise Set 4.5 274 4.6 Application: Computer Graphics 276 Graphics Operations in R superscript 2 277 Projection 286 Exercise Set 4.6 289 Review Exercises for Chapter 4 291 Chapter 4: Chapter Test 293 Chapter 5: Eigenvalues and Eigenvectors 296 5.1 Eigenvalues and Eigenvectors 297 Geometric Interpretation of Eigenvalues and Eigenvectors 299 Eigenspaces 300 Eigenvalues and Eigenvectors of Linear Operators 304 Exercise Set 5.1 306 5.2 Diagonalization 308 Diagonalizable Linear Operators 316 Exercise Set 5.2 319 5.3 Application: Systems of Linear Differential Equations 321 Uncoupled Systems 321 The Phase Plane 322 Diagonalization 323 Exercise Set 5.3 330 5.4 Application: Markov Chains 331 State Vectors and Transition Matrices 332 Diagonalizing the Transition Matrix 333 Steady-State Vector 334 Exercise Set 5.4 336 Review Exercises for Chapter 5 337 Chapter 5: Chapter Test 339 Chapter 6: Inner Product Spaces 342 6.1 The Dot Product on R superscript n 344 The Angle between Two Vectors 348 Exercise Set 6.1 352 6.2 Inner Product Spaces 354 Orthogonal Sets 358 Exercise Set 6.2 362 6.3 Orthonormal Bases 363 Orthogonal Projections 364 Gram-Schmidt Process 368 A Geometric Interpretation of the Gram-Schmidt Process 369 Exercise Set 6.3 373 6.4 Orthogonal Complements 376 Matrices 383 Linear Systems 384 Exercise Set 6.4 385 6.5 Application: Least Squares Approximation 387 Least Squares Solutions 390 Linear Regression 392 Fourier Polynomials 394 Exercise Set 6.5 396 6.6 Diagonalization of Symmetric Matrices 398 Orthogonal Diagonalization 400 Exercise Set 6.6 404 6.7 Application: Quadratic Forms 406 Rotation of Axes 406 Quadric Surfaces 412 Exercise Set 6.7 413 6.8 Application: Singular Value Decomposition 413 Singular Values of an m × n Matrix 414 Singular Value Decomposition (SVD) 417 The Four Fundamental Subspaces 422 Data Compression 422 Exercise Set 6.8 424 Review Exercises for Chapter 6 425 Chapter 6: Chapter Test 427 Appendix A: Preliminaries 430 A.1 Algebra of Sets 430 Operations on Sets 431 Exercise Set A.1 435 A.2 Functions 436 Inverse Functions 439 Composition of Functions 441 Exercise Set A.2 444 A.3 Techniques of Proof 445 Direct Argument 446 Contrapositive Argument 447 Contradiction Argument 447 Quantifiers 448 Exercise Set A.3 449 A.4 Mathematical Induction 450 Binomial Coefficients and the Binomial Theorem 456 Exercise Set A.4 459 Answers to Odd-Numbered Exercises 461 Index 500 Over the last few decades, linear algebra has become more relevant than ever. Applications have increased not only in quantity but also in diversity, with linear systems being used to solve problems in chemistry, engineering, economics, nutrition, urban planning, and more. DeFranza and Gagliardi introduce students to the topic in a clear, engaging, and easy-to-follow manner. Topics are developed fully before moving on to the next through a series of natural connections. The result is a solid introduction to linear algebra for undergraduates' first course. **__Outstanding features include:__** Early coverage of vector spaces, providing the abstract theory necessary to understand applications Exercises that range from routine to more challenging, extending the concepts and techniques by asking students to construct complete arguments Numerous examples designed to develop intuition and prepare readers to think conceptually about topics as they are introduced Fact summaries to end each chapter that use nontechnical language to recapitulate details and formulas Not-for-sale instructor resource material available to college and university faculty only; contact publisher directly. **Brief Table of Contents** 1. Systems of Linear Equations and Matrices 2. Linear Combinations and Linear Independence 3. Vector Spaces 4. Linear Transformations 5. Eigenvalues and Eigenvectors 6. Inner Product Spaces