Algebra Books

Book SynopsisThis upper undergraduate abstract algebra text covers classical themes on groups, rings and fields in depth, augmented with a strong emphasis on irreducible polynomials, a fresh approach to modules and linear algebra, a fresh take on Gröbner theory, and a group theoretic treatment of Rejewski's deciphering of the Enigma machine.Trade Review'This is a very good book, which provides an excellent introduction to modern algebra for senior undergraduate or beginning graduate students. The book includes a thorough coverage of the standard topics in the theories of groups, rings, fields, modules and Galois theory, taking a conceptual approach to algebra. For instance, the group theory part focuses on group actions, the ring theory exposition very appropriately stresses unique factorization properties, and the Galois theory part details some rather conceptual applications. Some of the less standard, very interesting topics are also present, including the breaking of the Enigma machine, as well as an in-depth look at division algorithms, including Gröbner bases. The book includes numerous exercises. All in all, a great new algebra text!' Lenny Fukshansky, Claremont McKenna College'An excellent textbook for an advanced undergraduate or a beginning graduate course on abstract algebra. Includes a lucid discussion of all core topics in group theory, commutative ring theory, Galois theory, and modules over principal ideal domains. I would describe this book as a simplified version of the classical textbook by Dummit and Foote.' Mihran Papikian, Pennsylvania State University'The 'comprehensive' in the title is no joke: this book walks the reader through a sea of detailed examples and computations in abstract algebra. These, and the exercises, are well thought out and will appeal to the student who likes a very hands-on kind of textbook. The group actions and division algorithms chapters are my personal favourites, as the computational nature of those topics plays to the strengths of the authors.' Nick Gurski, Case Western Reserve University'This is a great introduction to abstract algebra for graduate students and mathematically mature undergraduates.' Thomas Garrity, Williams College'Lawrence and Zorzitto's treatment of Abstract Algebra is lucid and thorough. I am particularly pleased to see the inclusion of Gröbner basis theory in a way that is accessible to introductory students, as it makes possible the exploration of polynomial ideals to great depth.' Jeffrey Clark, Elon UniversityTable of ContentsContents; Preface; 1. A refresher on the integers; 2. A first look at groups; 3. Groups acting on sets; 4. Basics on rings-mostly commutative; 5. Primes and unique factorization; 6. Algebraic field extensions; 7. Applications of galois theory; 8. Modules over principal ideal domains; 9. Division algorithms; Appendix A: Infinite sets.

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Book SynopsisThe grade-saving Algebra I companion, with hundreds of additional practice problems online Algebra I Workbook For Dummies is your solution to the Algebra brain-block. With hundreds of practice and example problems mapped to the typical high school Algebra class, you''ll crack the code in no time! Each problem includes a full explanation so you can see where you went wrongor rightevery step of the way. From fractions to FOIL and everything in between, this guide will help you grasp the fundamental concepts you''ll use in every other math class you''ll ever take. This new third edition includes access to an online test bank, where you''ll find bonus chapter quizzes to help you test your understanding and pinpoint areas in need of review. Whether you''re preparing for an exam or seeking a start-to-finish study aid, this workbook is your ticket to acing algebra. Master basic operations and properties to solve any problem Simplify expressions witTable of ContentsIntroduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 3 Beyond the Book 3 Where to Go from Here 4 Part 1: Getting Down to the Nitty-Gritty on Basic Operations 5 Chapter 1: Deciphering Signs in Numbers 7 Assigning Numbers Their Place 7 Reading and Writing Absolute Value 9 Adding Signed Numbers 10 Making a Difference with Signed Numbers 11 Multiplying Signed Numbers 12 Dividing Signed Numbers 14 Answers to Problems on Signed Numbers 15 Chapter 2: Incorporating Algebraic Properties 17 Getting a Grip on Grouping Symbols 17 Distributing the Wealth 19 Making Associations Work 20 Computing by Commuting 21 Answers to Problems on Algebraic Properties 23 Chapter 3: Making Fractions and Decimals Behave 25 Converting Improper and Mixed Fractions 25 Finding Fraction Equivalences 27 Making Proportional Statements 28 Finding Common Denominators 30 Adding and Subtracting Fractions 31 Multiplying and Dividing Fractions 32 Simplifying Complex Fractions 35 Changing Fractions to Decimals and Vice Versa 36 Performing Operations with Decimals 38 Answers to Problems on Fractions 39 Chapter 4: Exploring Exponents 45 Multiplying and Dividing Exponentials 45 Raising Powers to Powers 47 Using Negative Exponents 49 Writing Numbers with Scientific Notation 50 Answers to Problems on Discovering Exponents 52 Chapter 5: Taming Rampaging Radicals 55 Simplifying Radical Expressions 55 Rationalizing Fractions 57 Arranging Radicals as Exponential Terms 58 Using Fractional Exponents 60 Simplifying Expressions with Exponents 61 Estimating Answers 63 Answers to Problems on Radicals 64 Chapter 6: Simplifying Algebraic Expressions 67 Adding and Subtracting Like Terms 68 Multiplying and Dividing Algebraically 69 Incorporating Order of Operations 70 Evaluating Expressions 71 Answers to Problems on Algebraic Expressions 74 Part 2: Changing the Format of Expressions 77 Chapter 7: Specializing in Multiplication Matters 79 Distributing One Factor over Many 79 Curses, FOILed again — or not 80 Squaring binomials 82 Multiplying the sum and difference of the same two terms 83 Cubing binomials 84 Creating the Sum and Difference of Cubes 85 Raising binomials to higher powers 86 Answers to Problems on Multiplying Expressions 88 Chapter 8: Dividing the Long Way to Simplify Algebraic Expressions 91 Dividing by a Monomial 91 Dividing by a Binomial 93 Dividing by Polynomials with More Terms 96 Simplifying Division Synthetically 97 Answers to Problems on Division 99 Chapter 9: Figuring on Factoring 103 Pouring Over Prime Factorizations 103 Factoring Out the Greatest Common Factor 105 Reducing Algebraic Fractions 106 Answers to Problems on Factoring Expressions 108 Chapter 10: Taking the Bite Out of Binomial Factoring 111 Factoring the Difference of Squares 112 Factoring Differences and Sums of Cubes 113 Making Factoring a Multiple Mission 114 Answers to Problems on Factoring 115 Chapter 11: Factoring Trinomials and Special Polynomials 117 Focusing First on the Greatest Common Factor (GCF) 118 “Un”wrapping the FOIL 119 Factoring Quadratic-Like Trinomials 121 Factoring Trinomials Using More than One Method 122 Factoring by Grouping .124 Putting All the Factoring Together 126 Answers to Problems on Factoring Trinomials and Other Expressions 128 Part 3: Seek and Ye Shall Find Solutions 131 Chapter 12: Lining Up Linear Equations 133 Using the Addition/Subtraction Property 133 Using the Multiplication/Division Property 135 Putting Several Operations Together 136 Solving Linear Equations with Grouping Symbols 138 Working It Out with Fractions 140 Solving Proportions 142 Answers to Problems on Solving Linear Equations 144 Chapter 13: Muscling Up to Quadratic Equations 151 Using the Square Root Rule 152 Solving by Factoring 153 Using the Quadratic Formula 155 Completing the Square 158 Dealing with Impossible Answers 159 Answers to Problems on Solving Quadratic Equations 161 Chapter 14: Yielding to Higher Powers 167 Determining How Many Possible Roots 168 Applying the Rational Root Theorem 169 Using the Factor/Root Theorem 170 Solving by Factoring 172 Solving Powers That Are Quadratic-Like 174 Answers to Problems on Solving Higher Power Equations 176 Chapter 15: Reeling in Radical and Absolute Value Equations 179 Squaring Both Sides to Solve Radical Equations 180 Doubling the Fun with Radical Equations .182 Solving Absolute Value Equations 183 Answers to Problems on Radical and Absolute Value Equations 185 Chapter 16: Getting Even with Inequalities 189 Using the Rules to Work on Inequality Statements 190 Rewriting Inequalities by Using Interval Notation 191 Solving Linear Inequalities 192 Solving Quadratic Inequalities 193 Dealing with Polynomial and Rational Inequalities 195 Solving Absolute Value Inequalities 196 Solving Complex Inequalities 198 Answers to Problems on Working with Inequalities 199 Part 4: Solving Story Problems and Sketching Graphs 203 Chapter 17: Facing Up to Formulas 205 Working with Formulas 206 Deciphering Perimeter, Area, and Volume 207 Using perimeter formulas to get around 207 Squaring off with area formulas 209 Working with volume formulas 211 Getting Interested in Using Percent 213 Answers to Problems on Using Formulas 215 Chapter 18: Making Formulas Work in Basic Story Problems 219 Applying the Pythagorean Theorem 220 Using Geometry to Solve Story Problems 221 Putting Distance, Rate, and Time in a Formula 224 Examining the distance-rate-time formula 224 Going the distance with story problems 226 Answers to Making Formulas Work in Basic Story Problems 228 Chapter 19: Relating Values in Story Problems 233 Tackling Age Problems 234 Tackling Consecutive Integer Problems 235 Working Together on Work Problems 238 Answers to Relating Values in Story Problems 240 Chapter 20: Measuring Up with Quality and Quantity Story Problems 243 Achieving the Right Blend with Mixtures Problems 244 Concocting the Correct Solution One Hundred Percent of the Time 246 Dealing with Money Problems 248 Answers to Problems on Measuring Up with Quality and Quantity 250 Chapter 21: Getting a Handle on Graphing 255 Thickening the Plot with Points 255 Sectioning Off by Quadrants 257 Using Points to Lay Out Lines 258 Graphing Lines with Intercepts 260 Computing Slopes of Lines 261 Graphing with the Slope-Intercept Form 263 Changing to the Slope-Intercept Form 265 Writing Equations of Lines 266 Picking on Parallel and Perpendicular Lines 267 Finding Distances between Points 268 Finding the Intersections of Lines 269 Graphing Parabolas and Circles 270 Graphing with Transformations 272 Answers to Problems on Graphing 275 Part 5: The Part of Tens 283 Chapter 22: Ten Common Errors That Get Noticed 285 Squaring a Negative or Negative of a Square 285 Squaring a Binomial 286 Operating on Radicals 286 Distributing a Negative Throughout 287 Fracturing Fractions 287 Raising a Power to a Power 288 Making Negative Exponents Flip 288 Making Sense of Reversing the Sense 288 Using the Slope Formula Correctly 289 Writing Several Fractions as One 289 Chapter 23: Ten Quick Tips to Make Algebra a Breeze 291 Flipping Proportions 291 Multiplying Through to Get Rid of Fractions 292 Zeroing In on Fractions 292 Finding a Common Denominator 292 Dividing by 3 or 9 293 Dividing by 2, 4, or 8 293 Commuting Back and Forth 293 Factoring Quadratics 294 Making Radicals Less Rad, Baby 294 Applying Acronyms 294 Index 295

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Book SynopsisTable of ContentsPreface xiii Part One — Matrices 1 Basic properties of vectors and matrices 3 1 Introduction 3 2 Sets 3 3 Matrices: addition and multiplication 4 4 The transpose of a matrix 6 5 Square matrices 6 6 Linear forms and quadratic forms 7 7 The rank of a matrix 9 8 The inverse 10 9 The determinant 10 10 The trace 11 11 Partitioned matrices 12 12 Complex matrices 14 13 Eigenvalues and eigenvectors 14 14 Schur’s decomposition theorem 17 15 The Jordan decomposition 18 16 The singular-value decomposition 20 17 Further results concerning eigenvalues 20 18 Positive (semi)definite matrices 23 19 Three further results for positive definite matrices 25 20 A useful result 26 21 Symmetric matrix functions 27 Miscellaneous exercises 28 Bibliographical notes 30 2 Kronecker products, vec operator, and Moore-Penrose inverse 31 1 Introduction 31 2 The Kronecker product 31 3 Eigenvalues of a Kronecker product 33 4 The vec operator 34 5 The Moore-Penrose (MP) inverse 36 6 Existence and uniqueness of the MP inverse 37 7 Some properties of the MP inverse 38 8 Further properties 39 9 The solution of linear equation systems 41 Miscellaneous exercises 43 Bibliographical notes 45 3 Miscellaneous matrix results 47 1 Introduction 47 2 The adjoint matrix 47 3 Proof of Theorem 3.1 49 4 Bordered determinants 51 5 The matrix equation AX = 0 51 6 The Hadamard product 52 7 The commutation matrix Kmn 54 8 The duplication matrix Dn 56 9 Relationship between Dn+1 and Dn, I 58 10 Relationship between Dn+1 and Dn, II 59 11 Conditions for a quadratic form to be positive (negative) subject to linear constraints 60 12 Necessary and sufficient conditions for r(A : B) = r(A) + r(B) 63 13 The bordered Gramian matrix 65 14 The equations X1A + X2B′ = G1,X1B = G2 67 Miscellaneous exercises 69 Bibliographical notes 70 Part Two — Differentials: the theory 4 Mathematical preliminaries 73 1 Introduction 73 2 Interior points and accumulation points 73 3 Open and closed sets 75 4 The Bolzano-Weierstrass theorem 77 5 Functions 78 6 The limit of a function 79 7 Continuous functions and compactness 80 8 Convex sets 81 9 Convex and concave functions 83 Bibliographical notes 86 5 Differentials and differentiability 87 1 Introduction 87 2 Continuity 88 3 Differentiability and linear approximation 90 4 The differential of a vector function 91 5 Uniqueness of the differential 93 6 Continuity of differentiable functions 94 7 Partial derivatives 95 8 The first identification theorem 96 9 Existence of the differential, I 97 10 Existence of the differential, II 99 11 Continuous differentiability 100 12 The chain rule 100 13 Cauchy invariance 102 14 The mean-value theorem for real-valued functions 103 15 Differentiable matrix functions 104 16 Some remarks on notation 106 17 Complex differentiation 108 Miscellaneous exercises 110 Bibliographical notes 110 6 The second differential 111 1 Introduction 111 2 Second-order partial derivatives 111 3 The Hessian matrix 112 4 Twice differentiability and second-order approximation, I 113 5 Definition of twice differentiability 114 6 The second differential 115 7 Symmetry of the Hessian matrix 117 8 The second identification theorem 119 9 Twice differentiability and second-order approximation, II 119 10 Chain rule for Hessian matrices 121 11 The analog for second differentials 123 12 Taylor’s theorem for real-valued functions 124 13 Higher-order differentials 125 14 Real analytic functions 125 15 Twice differentiable matrix functions 126 Bibliographical notes 127 7 Static optimization 129 1 Introduction 129 2 Unconstrained optimization 130 3 The existence of absolute extrema 131 4 Necessary conditions for a local minimum 132 5 Sufficient conditions for a local minimum: first-derivative test 134 6 Sufficient conditions for a local minimum: second-derivative test 136 7 Characterization of differentiable convex functions 138 8 Characterization of twice differentiable convex functions 141 9 Sufficient conditions for an absolute minimum 142 10 Monotonic transformations 143 11 Optimization subject to constraints 144 12 Necessary conditions for a local minimum under constraints 145 13 Sufficient conditions for a local minimum under constraints 149 14 Sufficient conditions for an absolute minimum under constraints 154 15 A note on constraints in matrix form 155 16 Economic interpretation of Lagrange multipliers 155 Appendix: the implicit function theorem 157 Bibliographical notes 159 Part Three — Differentials: the practice 8 Some important differentials 163 1 Introduction 163 2 Fundamental rules of differential calculus 163 3 The differential of a determinant 165 4 The differential of an inverse 168 5 Differential of the Moore-Penrose inverse 169 6 The differential of the adjoint matrix 172 7 On differentiating eigenvalues and eigenvectors 174 8 The continuity of eigenprojections 176 9 The differential of eigenvalues and eigenvectors: symmetric case 180 10 Two alternative expressions for dλ 183 11 Second differential of the eigenvalue function 185 Miscellaneous exercises 186 Bibliographical notes 189 9 First-order differentials and Jacobian matrices 191 1 Introduction 191 2 Classification 192 3 Derisatives 192 4 Derivatives 194 5 Identification of Jacobian matrices 196 6 The first identification table 197 7 Partitioning of the derivative 197 8 Scalar functions of a scalar 198 9 Scalar functions of a vector 198 10 Scalar functions of a matrix, I: trace 199 11 Scalar functions of a matrix, II: determinant 201 12 Scalar functions of a matrix, III: eigenvalue 202 13 Two examples of vector functions 203 14 Matrix functions 204 15 Kronecker products 206 16 Some other problems 208 17 Jacobians of transformations 209 Bibliographical notes 210 10 Second-order differentials and Hessian matrices 211 1 Introduction 211 2 The second identification table 211 3 Linear and quadratic forms 212 4 A useful theorem 213 5 The determinant function 214 6 The eigenvalue function 215 7 Other examples 215 8 Composite functions 217 9 The eigenvector function 218 10 Hessian of matrix functions, I 219 11 Hessian of matrix functions, II 219 Miscellaneous exercises 220 Part Four — Inequalities 11 Inequalities 225 1 Introduction 225 2 The Cauchy-Schwarz inequality 226 3 Matrix analogs of the Cauchy-Schwarz inequality 227 4 The theorem of the arithmetic and geometric means 228 5 The Rayleigh quotient 230 6 Concavity of λ1 and convexity of λn 232 7 Variational description of eigenvalues 232 8 Fischer’s min-max theorem 234 9 Monotonicity of the eigenvalues 236 10 The Poincar´e separation theorem 236 11 Two corollaries of Poincar´e’s theorem 237 12 Further consequences of the Poincar´e theorem 238 13 Multiplicative version 239 14 The maximum of a bilinear form 241 15 Hadamard’s inequality 242 16 An interlude: Karamata’s inequality 242 17 Karamata’s inequality and eigenvalues 244 18 An inequality concerning positive semidefinite matrices 245 19 A representation theorem for ( ∑api )1/p 246 20 A representation theorem for (trAp)1/p 247 21 Hölder’s inequality 248 22 Concavity of log|A| 250 23 Minkowski’s inequality 251 24 Quasilinear representation of |A|1/n 253 25 Minkowski’s determinant theorem 255 26 Weighted means of order p 256 27 Schlömilch’s inequality 258 28 Curvature properties of Mp(x, a) 259 29 Least squares 260 30 Generalized least squares 261 31 Restricted least squares 262 32 Restricted least squares: matrix version 264 Miscellaneous exercises 265 Bibliographical notes 269 Part Five — The linear model 12 Statistical preliminaries 273 1 Introduction 273 2 The cumulative distribution function 273 3 The joint density function 274 4 Expectations 274 5 Variance and covariance 275 6 Independence of two random variables 277 7 Independence of n random variables 279 8 Sampling 279 9 The one-dimensional normal distribution 279 10 The multivariate normal distribution 280 11 Estimation 282 Miscellaneous exercises 282 Bibliographical notes 283 13 The linear regression model 285 1 Introduction 285 2 Affine minimum-trace unbiased estimation 286 3 The Gauss-Markov theorem 287 4 The method of least squares 290 5 Aitken’s theorem 291 6 Multicollinearity 293 7 Estimable functions 295 8 Linear constraints: the case M(R′) ⊂M(X′) 296 9 Linear constraints: the general case 300 10 Linear constraints: the case M(R′) ∩M(X′) = {0} 302 11 A singular variance matrix: the case M(X) ⊂M(V ) 304 12 A singular variance matrix: the case r(X′V +X) = r(X) 305 13 A singular variance matrix: the general case, I 307 14 Explicit and implicit linear constraints 307 15 The general linear model, I 310 16 A singular variance matrix: the general case, II 311 17 The general linear model, II 314 18 Generalized least squares 315 19 Restricted least squares 316 Miscellaneous exercises 318 Bibliographical notes 319 14 Further topics in the linear model 321 1 Introduction 321 2 Best quadratic unbiased estimation of σ2 322 3 The best quadratic and positive unbiased estimator of σ2 322 4 The best quadratic unbiased estimator of σ2 324 5 Best quadratic invariant estimation of σ2 326 6 The best quadratic and positive invariant estimator of σ2 327 7 The best quadratic invariant estimator of σ2 329 8 Best quadratic unbiased estimation: multivariate normal case 330 9 Bounds for the bias of the least-squares estimator of σ2, I 332 10 Bounds for the bias of the least-squares estimator of σ2, II 333 11 The prediction of disturbances 335 12 Best linear unbiased predictors with scalar variance matrix 336 13 Best linear unbiased predictors with fixed variance matrix, I 338 14 Best linear unbiased predictors with fixed variance matrix, II 340 15 Local sensitivity of the posterior mean 341 16 Local sensitivity of the posterior precision 342 Bibliographical notes 344 Part Six — Applications to maximum likelihood estimation 15 Maximum likelihood estimation 347 1 Introduction 347 2 The method of maximum likelihood (ML) 347 3 ML estimation of the multivariate normal distribution 348 4 Symmetry: implicit versus explicit treatment 350 5 The treatment of positive definiteness 351 6 The information matrix 352 7 ML estimation of the multivariate normal distribution: distinct means 354 8 The multivariate linear regression model 354 9 The errors-in-variables model 357 10 The nonlinear regression model with normal errors 359 11 Special case: functional independence of mean and variance parameters 361 12 Generalization of Theorem 15.6 362 Miscellaneous exercises 364 Bibliographical notes 365 16 Simultaneous equations 367 1 Introduction 367 2 The simultaneous equations model 367 3 The identification problem 369 4 Identification with linear constraints on B and Γ only 371 5 Identification with linear constraints on B, Γ, and ∑ 371 6 Nonlinear constraints 373 7 FIML: the information matrix (general case) 374 8 FIML: asymptotic variance matrix (special case) 376 9 LIML: first-order conditions 378 10 LIML: information matrix 381 11 LIML: asymptotic variance matrix 383 Bibliographical notes 388 17 Topics in psychometrics 389 1 Introduction 389 2 Population principal components 390 3 Optimality of principal components 391 4 A related result 392 5 Sample principal components 393 6 Optimality of sample principal components 395 7 One-mode component analysis 395 8 One-mode component analysis and sample principal components 398 9 Two-mode component analysis 399 10 Multimode component analysis 400 11 Factor analysis 404 12 A zigzag routine 407 13 A Newton-Raphson routine 408 14 Kaiser’s varimax method 412 15 Canonical correlations and variates in the population 414 16 Correspondence analysis 417 17 Linear discriminant analysis 418 Bibliographical notes 419 Part Seven — Summary 18 Matrix calculus: the essentials 423 1 Introduction 423 2 Differentials 424 3 Vector calculus 426 4 Optimization 429 5 Least squares 431 6 Matrix calculus 432 7 Interlude on linear and quadratic forms 434 8 The second differential 434 9 Chain rule for second differentials 436 10 Four examples 438 11 The Kronecker product and vec operator 439 12 Identification 441 13 The commutation matrix 442 14 From second differential to Hessian 443 15 Symmetry and the duplication matrix 444 16 Maximum likelihood 445 Further reading 448 Bibliography 449 Index of symbols 467 Subject index 471

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Book SynopsisBoost your chances of scoring higher at Algebra II Algebra II introduces students to complex algebra concepts in preparation for trigonometry and calculus. In this new edition of Algebra II Workbook For Dummies, high school and college students will work through the types of Algebra II problems they''ll see in class, including systems of equations, matrices, graphs, and conic sections. Plus, the book now comes with free 1-year access to chapter quizzes online! A recent report by ACT shows that over a quarter of ACT-tested 2012 high school graduates did not meet any of the four college readiness benchmarks in mathematics, English, reading, and science. Algebra II Workbook For Dummies presents tricky topics in plain English and short lessons, with examples and practice at every step to help students master the essentials, setting them up for success with each new lesson. Tracks to a typical Algebra II class Can be used as a suppTable of ContentsIntroduction About This Book 1 Foolish Assumptions 2 Icons Used in This Book 2 Beyond the Book 3 Where to Go from Here 3 Part 1: Getting Started With Algebra II 5 Chapter 1: Going Beyond Beginning Algebra 7 Good Citizenship: Following the Order of Operations and Other Properties 7 Specializing in Products and FOIL 10 Variables on the Side: Solving Linear Equations 11 Dealing with Linear Absolute Value Equations 12 Greater Math Skills: Equalizing Linear Inequalities 14 Answers to Problems on Going Beyond Beginning Algebra 16 Chapter 2: Handling Quadratic (and Quadratic-Like) Equations and Inequalities 21 Finding Reasonable Solutions with Radicals 22 UnFOILed Again! Successfully Factoring for Solutions 23 Your Bag of Tricks: Factoring Multiple Ways 25 Keeping Your Act Together: Factoring by Grouping 26 Resorting to the Quadratic Formula 27 Solving Quadratics by Completing the Square 29 Working with Quadratic-Like Equations 30 Checking Out Quadratic Inequalities 32 Answers to Problems on Quadratic (and Quadratic-Like) Equations and Inequalities 34 Chapter 3: Rooting Out the Rational, the Radical, and the Negative 43 Doing Away with Denominators with an LCD 44 Simplifying and Solving Proportions 46 Wrangling with Radicals 48 Changing Negative Attitudes toward Negative Exponents 49 Divided Powers: Solving Equations with Fractional Exponents 51 Answers to Problems on Rooting Out the Rational, the Radical, and the Negative 53 Chapter 4: Graphing for the Good Life 61 Coordinating Axes, Coordinates of Points, and Quadrants 62 Crossing the Line: Using Intercepts and Symmetry to Graph 64 Graphing Lines Using Slope-Intercept and Standard Forms 67 Graphing Basic Polynomial Curves 69 Grappling with Radical and Absolute Value Functions 71 Enter the Machines: Using a Graphing Calculator 73 Answers to Problems on Graphing for the Good Life 77 Part 2: Functions 89 Chapter 5: Formulating Functions 91 Evaluating Functions 91 Determining the Domain and Range of a Function 93 Recognizing Even, Odd, and One-to-One Functions 94 Composing Functions and Simplifying the Difference Quotient 96 Solving for Inverse Functions 99 Answers to Problems on Formulating Functions 101 Chapter 6: Specializing in Quadratic Functions 107 Finding Intercepts and the Vertex of a Parabola 108 Applying Quadratics to Real-Life Situations 109 Graphing Parabolas 111 Answers to Problems on Quadratic Functions 113 Chapter 7: Plugging in Polynomials 119 Finding Basic Polynomial Intercepts 120 Digging up More-Difficult Polynomial Roots with Factoring 122 Determining Where a Function Is Positive or Negative 123 Graphing Polynomials 125 Possible Roots and Where to Find Them: The Rational Root Theorem and Descartes’s Rule 127 Getting Real Results with Synthetic Division and the Remainder Theorem 130 Connecting the Factor Theorem with a Polynomial’s Roots 132 Answers to Problems on Plugging in Polynomials 134 Chapter 8: Acting Rationally with Functions 143 Determining Domain and Intercepts of Rational Functions 144 Introducing Vertical and Horizontal Asymptotes 145 Getting a New Slant with Oblique Asymptotes 147 Removing Discontinuities 148 Going the Limit: Limits at a Number and Infinity 149 Graphing Rational Functions 151 Answers to Problems on Rational Functions 156 Chapter 9: Exposing Exponential and Logarithmic Functions 163 Evaluating e-Expressions and Powers of e 164 Solving Exponential Equations 165 Making Cents: Applying Compound Interest and Continuous Compounding 167 Checking out the Properties of Logarithms 169 Presto-Chango: Expanding and Contracting Expressions with Log Functions 171 Solving Logarithmic Equations 173 They Ought to Be in Pictures: Graphing Exponential and Logarithmic Functions 175 Answers to Problems on Exponential and Logarithmic Functions 179 Part 3: Conics And Systems Of Equations 189 Chapter 10: Any Way You Slice It: Conic Sections 191 Putting Equations of Parabolas in Standard Form 192 Shaping Up: Determining the Focus and Directrix of a Parabola 194 Back to the Drawing Board: Sketching Parabolas 196 Writing the Equations of Circles and Ellipses in Standard Form 198 Determining Foci and Vertices of Ellipses 201 Rounding Out Your Sketches: Circles and Ellipses 203 Hyperbola: Standard Equations and Foci 205 Determining the Asymptotes and Intercepts of Hyperbolas 206 Sketching the Hyperbola 208 Answers to Problems on Conic Sections 211 Chapter 11: Solving Systems of Linear Equations 221 Solving Two Linear Equations Algebraically 221 Using Cramer’s Rule to Defeat Unruly Fractions 223 A Third Variable: Upping the Systems to Three Linear Equations 225 A Line by Any Other Name: Writing Generalized Solution Rules 227 Decomposing Fractions Using Systems 229 Answers to Problems on Systems of Equations 231 Chapter 12: Solving Systems of Nonlinear Equations and Inequalities 237 Finding the Intersections of Lines and Parabolas 237 Crossing Curves: Finding the Intersections of Parabolas and Circles 239 Appealing to a Higher Power: Dealing with Exponential Systems 240 Solving Systems of Inequalities 242 Answers to Problems on Solving Systems of Nonlinear Equations and Inequalities 245 Part 4: Other Good Stuff: Lists, Arrays, And Imaginary Numbers 251 Chapter 13: Getting More Complex with Imaginary Numbers 253 Simplifying Powers of i 254 Not Quite Brain Surgery: Doing Operations on Complex Numbers 255 “Dividing” Complex Numbers with a Conjugate 256 Solving Equations with Complex Solutions 257 Answers to Problems on Imaginary Numbers 259 Chapter 14: Getting Squared Away with Matrices 263 Describing Dimensions and Types of Matrices 263 Adding, Subtracting, and Doing Scalar Multiplication on Matrices 265 Trying Times: Multiplying Matrices by Each Other 267 The Search for Identity: Finding Inverse Matrices 268 Using Matrices to Solve Systems of Equations 272 Answers to Problems on Matrices 274 Chapter 15: Going Out of Sequence with Sequences and Series 279 Writing the Terms of a Sequence 279 Differences and Multipliers: Working with Special Sequences 282 Backtracking: Constructing Recursively Defined Sequences 283 Using Summation Notation 284 Finding Sums with Special Series 286 Answers to Problems on Sequences and Series 289 Chapter 16: Everything You Ever Wanted to Know about Sets and Counting 293 Writing the Elements of a Set from Rules or Patterns 294 Get Together: Combining Sets with Unions, Intersections, and Complements 295 Multiplication Countdowns: Simplifying Factorial Expressions 297 Checking Your Options: Using the Multiplication Property 298 Counting on Permutations When Order Matters 300 Mixing It Up with Combinations 301 Raising Binomials to Powers: Investigating the Binomial Theorem 303 Answers to Problems on Sets and Counting 304 Part 5: The Part Of Tens 309 Chapter 17: Basic Graphs 311 Putting Polynomials in Their Place 311 Lining Up Front and Center 312 Being Absolutely Sure with Absolute Value 313 Graphing Reciprocals of x and x2 .313 Rooting Out Square Root and Cube Root .314 Growing Exponentially with a Graph 315 Logging In on Logarithmic Graphing 316 Chapter 18: Ten Special Sequences and Their Sums 317 Adding as Easy as One, Two, Three 317 Summing Up the Squares 318 Finding the Sum of the Cubes 318 Not Being at Odds with Summing Odd Numbers 319 Evening Things Out by Adding Up Even Numbers 319 Adding Everything Arithmetic 319 Geometrically Speaking 320 Easing into a Sum for e 320 Signing In on the Sine 321 Powering Up on Powers of 2 322 Adding Up Fractions with Multiples for Denominators 322 Index 323

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Book SynopsisPractice your way to a great grade in Algebra I Algebra I: 1001 Practice Problems For Dummies gives you 1,001 opportunities to practice solving problems on all the major topics in Algebra Iin the book and online! Get extra help with tricky subjects, solidify what you've already learned, and get in-depth walk-throughs for every problem with this useful book. These practice problems and detailed answer explanations will get you solving for x in no-time, no matter what your skill level. Thanks to Dummies, you have a resource to you put key concepts into practice. Work through practice problems on all Algebra I topics covered in classStep through detailed solutions for every problem to build your understandingAccess practice questions online to study anywhere, any timeImprove your grade and up your study game with practice, practice, practiceThe material presented in Algebra I: 1001 Practice Problems For Dummies is an excellent resource for students, as well as parents and tutors looking to help supplement classroom instruction. Algebra I: 1001 Practice Problems For Dummies (9781119883470) was previously published as 1,001 Algebra I Practice Problems For Dummies (9781118446713). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.Table of ContentsIntroduction 1 Part 1: The Questions 5 Chapter 1: Signing on with Signed Numbers 7 Chapter 2: Recognizing Algebraic Properties and Notation 13 Chapter 3: Working with Fractions and Decimals 17 Chapter 4: Making Exponential Expressions and Operations More Compatible 23 Chapter 5: Raking in Radicals 29 Chapter 6: Creating More User-Friendly Algebraic Expressions 35 Chapter 7: Multiplying by One or More Terms 41 Chapter 8: Dividing Algebraic Expressions 47 Chapter 9: Factoring Basics 53 Chapter 10: Factoring Binomials 57 Chapter 11: Factoring Quadratic Trinomials 61 Chapter 12: Other Factoring Techniques 65 Chapter 13: Solving Linear Equations 69 Chapter 14: Taking on Quadratic Equations 73 Chapter 15: Solving Polynomials with Powers Three and Higher 79 Chapter 16: Reining in Radical and Absolute Value Equations 83 Chapter 17: Making Inequalities More Fair 87 Chapter 18: Using Established Formulas 93 Chapter 19: Using Formulas in Geometric Story Problems 101 Chapter 20: Tackling Traditional Story Problems 107 Chapter 21: Graphing Basics 113 Chapter 22: Using the Algebra of Lines 119 Chapter 23: Other Graphing Topics 123 Part 2: The Answers 127 Chapter 24: The Answers 129 Index 443

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Book SynopsisTable of ContentsIntroduction 1 Part 1: The Questions 5 Chapter 1: The Big Four Operations 7 Chapter 2: Less than Zero: Working with Negative Numbers 11 Chapter 3: You’ve Got the Power: Powers and Roots 17 Chapter 4: Following Orders: Order of Operations 23 Chapter 5: Big Four Word Problems 29 Chapter 6: Divided We Stand 35 Chapter 7: Factors and Multiples 43 Chapter 8: Word Problems about Factors and Multiples 49 Chapter 9: Fractions 53 Chapter 10: Decimals 63 Chapter 11: Percents 69 Chapter 12: Ratios and Proportions 75 Chapter 13: Word Problems for Fractions, Decimals, and Percents 79 Chapter 14: Scientific Notation 87 Chapter 15: Weights and Measures 91 Chapter 16: Geometry 97 Chapter 17: Graphing 109 Chapter 18: Statistics and Probability 115 Chapter 19: Set Theory 123 Chapter 20: Algebraic Expressions 127 Chapter 21: Solving Algebraic Equations 133 Chapter 22: Solving Algebra Word Problems 139 Part 2: The Answers 143 Chapter 23: Answers 145 Index 407

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Book SynopsisTable of ContentsIntroduction 1 Part 1: The Questions 5 Chapter 1: Reviewing Algebra Basics 7 Chapter 2: Solving Quadratic Equations and Nonlinear Inequalities 13 Chapter 3: Solving Radical and Rational Equations 21 Chapter 4: Graphs and Equations of Lines 27 Chapter 5: Functions 33 Chapter 6: Quadratic Functions and Relations 39 Chapter 7: Polynomial Functions and Equations 45 Chapter 8: Rational Functions 51 Chapter 9: Exponential and Logarithmic Functions 57 Chapter 10: Conic Sections 65 Chapter 11: Systems of Linear Equations 73 Chapter 12: Systems of Nonlinear Equations and Inequalities 79 Chapter 13: Working with Complex Numbers 85 Chapter 14: Matrices 91 Chapter 15: Sequences and Series 97 Chapter 16: Sets 103 Chapter 17: Counting Techniques and Probability 109 Part 2: The Answers 117 Chapter 18: The Answers 119 Index 499

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Book SynopsisDavid C. Lay, University of MarylandCollege Park Steven R. Lay, Lee University Judi J. McDonald, Washington State University

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Book SynopsisTable of Contents Chapter 1: First-Order Differential Equations Chapter 2: Mathematical Models and Numerical Methods Chapter 3: Linear Systems and Matrices Chapter 4: Vector Spaces Chapter 5: Higher-Order Linear Differential Equations Chapter 6: Eigenvalues and Eigenvectors Chapter 7: Linear Systems of Differential Equations Chapter 8: Matrix Exponential Methods Chapter 9: Nonlinear Systems and Phenomena Chapter 10: Laplace Transform Methods Chapter 11: Power Series Methods Appendix A: Existence and Uniqueness of Solutions Appendix B: Theory of Determinants Answers to Selected Problems Index Download the detailed table of contents

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Book SynopsisLife Cycle assessment (LCA) is a tool for environmental decision-support in relation to products from the cradle to the grave. Until now, more emphasis has been put on the inclusion quantitative models and databases and on the design of guidebooks for applying LCA than on the integrative aspect of combining these models and data.Trade Review"Heijungs and Suh's Computational Structure of Life Cycles Assessment fills a gap in the methodological literature supporting life-cycle assessments (LCA). It provides a consistent approach, terminology, and notation previously lacking and only partially addressed by archival literature and standardization efforts. Much of the book focuses on the computational aspects of inventory analysis using linear algebra. The construction has atleast three advantages. First, the method is compatible with current inventory data collection and management practices…Second, the computational structure forces the practitioner to account for the full life cycle of material and energy flows and explicitly accounts for "complications"….Third, the matrix structure facilitates impact assessment and interpretation as currently applied by LCA practitioners…Even though the linear algebra concepts used are quite basic, the text is really designed for the atleast somewhat experienced LCA practitioner or the graduate student with some level of comfort in applying mathematical models to complex systems…Finally, the computational structure presented is complete in taking the practitioner from inventory analysis through interpretation.. Practitioners who read this text will benefit from the author’s experiences in applying LCA and developing LCA methods…" Journal of Industrial Ecology, 7:2 (2003)Table of ContentsPreface. 1. Introduction. 2. The basic model for inventory analysis. 3. The refined model for inventory analysis. 4. Advanced topics in inventory analysis*. 5. Relation with input-output analysis*. 6. Perturbation theory. 7. Structural theory. 8. Beyond the inventory analysis. 9. Further extensions*. 10. Issues of implementation*. A. Matrix algebra. B. Main terms and symbols. C. Matlab code for most important algorithms. References. Index.

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Book SynopsisThe most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature - from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world. With admirable clarity, two veteran math educators take us on a fascinating tour of the many ramifications of the Fibonacci numbers. They begin with a brief history of a distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples). In art, architecture, the stock market, and other areas of society and culture, they point out numerous examples of the Fibonacci sequence as well as its derivative, the "golden ratio." And of course in mathematics, as the authors amply demonstrate, there are almost boundless applications in probability, number theory, geometry, algebra, and Pascal's triangle, to name a few.Accessible and appealing to even the most math-phobic individual, this fun and enlightening book allows the reader to appreciate the elegance of mathematics and its amazing applications in both natural and cultural settings.Trade Review""This is a wonderful introduction...You may end up amazed and incredulous." - Leon M. Lederman, Nobel Laureate“The mathematics in this book is a delight: surprising, insightful, and comprehensive… the result is by turns rigorous, entertaining, and eye-poppingly speculative.”-New Scientist “…a work that, although aimed at a general audience and presupposing no knowledge of mathematics beyond the high school precalculus level, succeeds in entertaining all audiences…Educators, as well as the mathematically curious, are encouraged to pick up this volume. The discussions of Fibonacci numbers in nature, art, architecture, and music are very thorough…highly recommended.” -Choice“The authors have breathed life into what could be considered a fairly dry subject by demonstrating how commonplace items make use of the Fibonacci numbers…there is a great deal of math involved but taken step at a time, it is not that difficult to understand and this understanding leads to a an even greater appreciation of everything from a flower garden to classical music. Overall, this is an interesting if challenging read for the layperson and a gold mine for the mathematically inclined.”-Monsters and Critics “…the authors have presented a compelling and well-developed book, and one that might well make converts out of some hard-core math phobics…an elegant book that enhances their argument that mathematics is ‘the queen of sciences'.”-Education Update “…delightful…accessible to anyone who enjoys or enjoyed high school mathematics. Mathematics teachers from middle school through college will find this book fun to read and useful in the classroom. The authors consider more properties, relationships, and applications of the Fibonacci numbers than most other sources do…I enjoyed reading this book…a valuable addition to the mathematical literature.”-Mathematics Teacher

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Book SynopsisBasic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises. More exercises (of the kind a student may expect in examination papers) are grouped at the end of each section. The book covers the most important basics of any first course on linear algebra, explaining the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations and complex numbers. Linear equations are treated via Hermite normal forms which provides a successful and concrete explanation of the notion of linear independence. Another important highlight is the connection between linear mappings and matrices leading to the change of basis theorem which opens the door to the notion of similarity. This new and revised edition features additional exercises and coverage of Cramer's rule (omitted from the first edition). However, it is the new, extra chapter on computer assistance that will be of particular interest to readers: this will take the form of a tutorial on the use of the "LinearAlgebra" package in MAPLE 7 and will deal with all the aspects of linear algebra developed within the book.Trade ReviewFrom the reviews: "It embodies a beautiful, concise and precise treatment of the subject, with succint numerical and algebra worked examples at the right points... an excellent textbook which is also eminently suitable for self-study..." Zentralblatt MATH "This is the second edition of a text for first-year students which covers the main themes of linear algebra in a succinct and readable way. … The book is well-written, with a very high standard of proof-reading, and there are full answers to all the exercises … . It should be welcomed as an excellent introductory textbook which could be used either for self-study and to complement a course of lectures." (Gerry Leversha, The Mathematical Gazette, Vol. 87 (509), 2003)Table of ContentsPreface Forward The Algebra of Matrices Some Applications of Matrices Systems of Linear Equations Invertible Matrices Vector Spaces Linear Mappings The Matrix Connection Determinants Eigenvalues and Eigenvectors The Minimum Polynomial Computer Assistance Solutions to the Exercises index

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Book SynopsisThis volume collects chapters that examine representation theory as connected with affine Lie algebras and their quantum analogues, in celebration of the impact Vyjayanthi Chari has had on this area. The opening chapters are based on mini-courses given at the conference “Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification”, held on the occasion of Chari’s 60th birthday at the Catholic University of America in Washington D.C., June 2018. The chapters that follow present a broad view of the area, featuring surveys, original research, and an overview of Vyjayanthi Chari’s significant contributions. Written by distinguished experts in representation theory, a range of topics are covered, including: String diagrams and categorification Quantum affine algebras and cluster algebras Steinberg groups for Jordan pairs Dynamical quantum determinants and Pfaffians Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification will be an ideal resource for researchers in the fields of representation theory and mathematical physics.Table of ContentsPublications of Vyjayanthi Chari.- Students of Vyjayanthi Chari.- Part I: Courses.- String Diagrams and Categorification.- Quantum Affine Algebras and Cluster Algebras.- Part II: Surveys.- Work of Vyjayanthi Chari.- Steinberg Groups for Jordan Pairs - An Introduction with Open Problems.- On the Hecke-Algebraic Approach for General Linear Groups over a p-adic Field.- Part III: Papers.- Categorical Representations and Classical p-adic Groups.- Formulae of l-Divided Powers in Uq(sl2),II.- Longest Weyl Group Elements in Action.- Dual Kashiwara Functions for the B(∞) Crystal in the Bipartite Case.- Lusztig's t-Analogue of weight multiplicity via Crystals.- Conormal Varieties on the Cominuscule Grassmannian.- Evaluation Modules for Quantum Toroidal gln Algebras.- Dynamical Quantum Determinants and Pfaffians.

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Book SynopsisThis book provides a unique tour of university mathematics with the help of Python. Written in the spirit of mathematical exploration and investigation, the book enables students to utilise Python to enrich their understanding of mathematics through: Calculation: performing complex calculations and numerical simulations instantly Visualisation: demonstrating key theorems with graphs, interactive plots and animations Extension: using numerical findings as inspiration for making deeper, more general conjectures. This book is for all learners of mathematics, with the primary audience being mathematics undergraduates who are curious to see how Python can enhance their understanding of core university material. The topics chosen represent a mathematical overview of what students typically study in the first and second years at university, namely analysis, calculus, vector calculus and geometry, differential equations and dynamical systems, linear algebra, abstract algebra and number theory, probability and statistics. As such, it can also serve as a preview of university mathematics for high-school students. The prerequisites for reading the book are a familiarity with standard A-Level mathematics (or equivalent senior high-school curricula) and a willingness to learn programming. For mathematics lecturers and teachers, this book is a useful resource on how Python can be seamlessly incorporated into the mathematics syllabus, assuming only basic knowledge of programming.Table of Contents1 Analysis.- 2 Calculus.- 3 Vector Calculus and Geometry.- 4 Differential Equations and Dynamical Systems.- 5 Linear Algebra.- 6 Abstract Algebra and Number Theory.- 7 Probability.- 8 Statistics.- Appendix A: Python 101.

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Book SynopsisLinear Equations.- Matrix Algebra.- Determinants.- Vector Spaces.- Eigenvalues and Eigenvectors.- Orthogonality.

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Book SynopsisThis textbook provides an accessible account of the history of abstract algebra, tracing a range of topics in modern algebra and number theory back to their modest presence in the seventeenth and eighteenth centuries, and exploring the impact of ideas on the development of the subject. Beginning with Gauss’s theory of numbers and Galois’s ideas, the book progresses to Dedekind and Kronecker, Jordan and Klein, Steinitz, Hilbert, and Emmy Noether. Approaching mathematical topics from a historical perspective, the author explores quadratic forms, quadratic reciprocity, Fermat’s Last Theorem, cyclotomy, quintic equations, Galois theory, commutative rings, abstract fields, ideal theory, invariant theory, and group theory. Readers will learn what Galois accomplished, how difficult the proofs of his theorems were, and how important Camille Jordan and Felix Klein were in the eventual acceptance of Galois’s approach to the solution of equations. The book also describes the relationship between Kummer’s ideal numbers and Dedekind’s ideals, and discusses why Dedekind felt his solution to the divisor problem was better than Kummer’s. Designed for a course in the history of modern algebra, this book is aimed at undergraduate students with an introductory background in algebra but will also appeal to researchers with a general interest in the topic. With exercises at the end of each chapter and appendices providing material difficult to find elsewhere, this book is self-contained and therefore suitable for self-study.Trade Review“This volume is well written and nicely complements other works on the history of algebra. It can be recommended to all mathematicians and students of mathematics who want to understand how algebra turned into the rather abstract field it is today.” (C. Baxa, Monatshefte für Mathematik, Vol. 201 (4), August, 2023)“The book under review is an excellent contribution to the history of abstract algebra and the beginnings of algebraic number theory. I recommend it to everyone interested in the history of mathematics.” (Franz Lemmermeyer, zbMATH 1411.01005, 2019)“This is a nice book to have around; it reflects careful scholarship and is filled with interesting material. … there is much to like about this book. It is quite detailed, contains a lot of information, is meticulously researched, and has an extensive bibliography. Anyone interested in the history of mathematics, or abstract algebra, will want to make the acquaintance of this book.” (Mark Hunacek, MAA Reviews, June 24, 2019)Table of ContentsIntroduction.- 1 Simple quadratic forms.- 2 Fermat’s Last Theorem.- 3 Lagrange’s theory of quadratic forms.- 4 Gauss’s Disquisitiones Arithmeticae.- 5 Cyclotomy.- 6 Two of Gauss’s proofs of quadratic reciprocity.- 7 Dirichlet’s Lectures.- 8 Is the quintic unsolvable?.- 9 The unsolvability of the quintic.- 10 Galois’s theory.- 11 After Galois – Introduction.- 12 Revision and first assignment.- 13 Jordan’s Traité.- 14 Jordan and Klein.- 15 What is ‘Galois theory’?.- 16 Algebraic number theory: cyclotomy.- 17 Dedekind’s first theory of ideals.- 18 Dedekind’s later theory of ideals.- 19 Quadratic forms and ideals.- 20 Kronecker’s algebraic number theory.- 21 Revision and second assignment.- 22 Algebra at the end of the 19th century.- 23 The concept of an abstract field.- 24 Ideal theory.- 25 Invariant theory.- 26 Hilbert’s Zahlbericht.- 27 The rise of modern algebra – group theory.- 28 Emmy Noether.- 29 From Weber to van der Waerden.- 30 Revision and final assignment.- A Polynomial equations in the 18th Century.- B Gauss and composition of forms.- C Gauss on quadratic reciprocity.- D From Jordan’s Traité.- E Klein’s Erlanger Programm.- F From Dedekind’s 11th supplement.- G Subgroups of S4 and S5.- H Curves.- I Resultants.- Bibliography.- Index.

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Book SynopsisThis is a basic introduction to modern algebra, providing a solid understanding of the axiomatic treatment of groups and then rings, aiming to promote a feeling for the evolutionary and historical development of the subject. It includes problems and fully worked solutions, enabling readers to master the subject rather than simply observing it.Table of Contents1 Sets and Mappings.- 2 The Integers.- 3 Introduction to Rings.- 4 Introduction to Groups.- 5 Rings.- 6 Topics in Group Theory.- Hints to Solutions.- Suggestions for Further Study.

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Book SynopsisVolume 2 applies the linear algebra concepts presented in Volume 1 to optimization problems which frequently occur throughout machine learning. This book blends theory with practice by not only carefully discussing the mathematical under pinnings of each optimization technique but by applying these techniques to linear programming, support vector machines (SVM), principal component analysis (PCA), and ridge regression. Volume 2 begins by discussing preliminary concepts of optimization theory such as metric spaces, derivatives, and the Lagrange multiplier technique for finding extrema of real valued functions. The focus then shifts to the special case of optimizing a linear function over a region determined by affine constraints, namely linear programming. Highlights include careful derivations and applications of the simplex algorithm, the dual-simplex algorithm, and the primal-dual algorithm. The theoretical heart of this book is the mathematically rigorous presentation of various nonlinear optimization methods, including but not limited to gradient decent, the Karush-Kuhn-Tucker (KKT) conditions, Lagrangian duality, alternating direction method of multipliers (ADMM), and the kernel method. These methods are carefully applied to hard margin SVM, soft margin SVM, kernel PCA, ridge regression, lasso regression, and elastic-net regression. Matlab programs implementing these methods are included.

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Book SynopsisLinear algebra and matrix theory are among the most important and most frequently applied branches of mathematics. They are especially important in solving engineering and economic models, where either the model is assumed linear, or the nonlinear model is approximated by a linear model, and the resulting linear model is examined.This book is mainly a textbook, that covers a one semester upper division course or a two semester lower division course on the subject.The second edition will be an extended and modernized version of the first edition. We added some new theoretical topics and some new applications from fields other than economics. We also added more difficult exercises at the end of each chapter which require deep understanding of the theoretical issues. We also modernized some proofs in the theoretical discussions which give better overview of the study material. In preparing the manuscript we also corrected the typos and errors, so the second edition will be a corrected, extended and modernized new version of the first edition.

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Book SynopsisThis is the first volume of the two-volume book on linear algebra, in the University of Tokyo (UTokyo) Engineering Course.The objective of this volume is to present, from the engineering viewpoint, the standard mathematical results in linear algebra such as those on systems of equations and eigenvalue problems. In addition to giving mathematical theorems and formulas, it explains how the mathematical concepts such as rank, eigenvalues, and singular values are linked to engineering applications and numerical computations.In particular, the following four aspects are emphasized.

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Book SynopsisIn China, lots of excellent maths students take an active interest in various maths contests and the best six senior high school students will be selected to form the IMO National Team to compete in the International Mathematical Olympiad. In the past ten years China's IMO Team has achieved outstanding results — they won the first place almost every year.The authors are coaches of China's IMO National Team, whose students have won many gold medals many times in IMO.This book is part of the Mathematical Olympiad Series which discusses several aspects related to maths contests, such as algebra, number theory, combinatorics, graph theory and geometry. The book explains many basic techniques for proving inequalities such as direct comparison, method of magnifying and reducing, substitution method, construction method, and so on.

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Book SynopsisAlgebra is like a foreign language for many students. They have difficulty translating the words, their definitions and how it applies to problem-solving. Tussy/Gustafson''s ELEMENTARY AND INTERMEDIATE ALGEBRA, 6th Edition, addresses these concerns, giving you the tools needed to understand the language of algebra. Strategy and Why explanations in the worked examples show the how and the why behind problem-solving. Algebra is not just about the x -- it''s also about the WHY. The text contains many opportunities to apply the algebraic skills you have learned to solve a wide variety of interesting real-life applications using a six-step problem-solving strategy. In combination, the text and WebAssign will guide you through an integrated learning process that will expand your reasoning abilities as it teaches you how to read, write and think mathematically using the language of Algebra.

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Book SynopsisReveals the hidden connections discovered over the last half-century that explain the existence of these mysterious mathematical objects. The book aims to convince the reader that the underlying algebro-geometric structure of soliton equations provides an elegant explanation of something seemingly miraculous.Trade ReviewThis book challenges and intrigues from beginning to end. It would be a treat to use for a capstone course or senior seminar." —William J. Satzer, MAA Reviews on Glimpses of Soliton Theory (First Edition)Table of Contents Differential equations Developing PDE intuition The story of solitons Elliptic curves and KdV traveling waves KdV $n$-solitons and $\tau$-functions Multiplying and factoring differential operators Eigenfunctions and isospectrality Lax form for KdV and other soliton equations The KP equation and bilinear KP equation $\Gamma_{2,4}$ and the bilinear KP equation Pseudo-differential operators and the KP hierarchy $\Gamma{k,n}$ and the bilinear KP hierarchy Concluding remarks Mathematica guide Complex numbers Ideas for independent projects References Glossary of symbols Index

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Book SynopsisThis textbook provides a modern introduction to linear algebra, a mathematical discipline every first year undergraduate student in physics and engineering must learn. A rigorous introduction into the mathematics is combined with many examples, solved problems, and exercises as well as scientific applications of linear algebra. These include applications to contemporary topics such as internet search, artificial intelligence, neural networks, and quantum computing, as well as a number of more advanced topics, such as Jordan normal form, singular value decomposition, and tensors, which will make it a useful reference for a more experienced practitioner. Structured into 27 chapters, it is designed as a basis for a lecture course and combines a rigorous mathematical development of the subject with a range of concisely presented scientific applications. The main text contains many examples and solved problems to help the reader develop a working knowledge of the subject and every chapter comes with exercises.Trade ReviewThe authors are uniquely well qualified to produce a textbook suitable for first-year university students. * David Matravers, University of Portsmouth *Linear Algebra is a core undergraduate course not only in Mathematics but also in Physics, Chemistry, Biology and Computer Science. This textbook brilliantly succeeds in catering to such a wide audience by covering a broad range of formal developments along with concrete applications and is unique in its presentation of the topic. * Richard Joseph Szabo, Heriot-Watt University *Lukas has written an impressive mathematical textbook that covers standard introductory linear algebra topics along with advanced concepts that will appeal to many readers. * Choice *Table of Contents1: Linearity - an informal introduction 2: Sets and functions 3: Groups 4: Fields 5: Coordinate vectors 6: Vector spaces 7: Elementary vector space properties 8: Vector subspaces 9: The dot product 10: Vector and triple product 11: Lines and planes 12: Introduction to linear maps 13: Matrices 14: The structure of linear maps 15: Linear maps in terms of matrices 16: Computing with matrices 17: Linear systems 18: Determinants 19: Basics of eigenvalues 20: Diagonalising linear maps 21: The Jordan normal form 22: Scalar products 23: Adjoint and unitary maps 24: Diagonalisation - again 25: Bi-linear and sesqui-linear forms 26: The dual vector space 27: Tensors

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Book SynopsisGraduate mathematics students will find this book an easy-to-follow, step-by-step guide to the subject. Rotman’s book gives a treatment of homological algebra which approaches the subject in terms of its origins in algebraic topology.Trade ReviewFrom the reviews of the second edition:"Joseph J. Rotman is a renowned textbook author in contemporary mathematics. Over the past four decades, he has published numerous successful texts of introductory character, mainly in the field of modern abstract algebra and its related disciplines. … Now, in the current second edition, the author has reworked the original text considerably. While the first edition covered exclusively aspects of the homological algebra of groups, rings, and modules, that is, topics from its first period of development, the new edition includes some additional material from the second period, together with numerous other, more recent results from the homological algebra of groups, rings, and modules. The new edition has almost doubled in size and represents a substantial updating of the classic original. … All together, a popular classic has been turned into a new, much more topical and comprehensive textbook on homological algebra, with all the great features that once distinguished the original, very much to the belief [of its] new generation of readers." (Werner Kleinert, Zentralblatt)"The new expanded second edition … attempts to cover more ground, basically going from the (concrete) category of modules over a given ring, as in the first edition, to an abelian category and to treat the important example of the category of sheaves on a topological space. … the exercise at the end of every section, plenty of examples and motivation for the many new concepts set this book apart and make it an ideal textbook for a course on the subject." (Felipe Zaldivar, MAA Online, December, 2008)"This is the second edition of Rotman’s introduction to the more classical aspects of homological algebra … . The book is mainly concerned with homological algebra in module categories … . The book is full of illustrative examples and exercises. It contains many references for further study and also to original sources. All this makes Rotman’s book very convenient for beginners in homological algebra as well as a reference book." (Fernando Muro, Mathematical Reviews, Issue 2009 i)Table of ContentsHom and Tensor.- Special Modules.- Specific Rings.- Setting the Stage.- Homology.- Tor and Ext.- Homology and Rings.- Homology and Groups.- Spectral Sequences.

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Book SynopsisIt then proceeds to a discussion of modules, emphasizing a comparison with vector spaces, and presents a thorough discussion of inner product spaces, eigenvalues, eigenvectors, and finite dimensional spectral theory, culminating in the finite dimensional spectral theorem for normal operators.Trade ReviewFrom the reviews of the first edition:"… The book is very well written and has a good set of exercises. It is a suitable choice as a graduate textbook as well as a reference book." A.A. Jafarian for ZentralblattMATHFrom the reviews of the second edition:"In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. … As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. … the exercises are rewritten and expanded. … Overall, I found the book a very useful one. … It is a suitable choice as a graduate text or as a reference book." (Ali-Akbar Jafarian, Zentralblatt MATH, Vol. 1085, 2006)"This is a formidable volume, a compendium of linear algebra theory, classical and modern … . The development of the subject is elegant … . The proofs are neat … . The exercise sets are good, with occasional hints given for the solution of trickier problems. … It represents linear algebra and does so comprehensively." (Henry Ricardo, MathDL, May, 2005)From the reviews of the third edition:“This is the 3rd edition of a well written graduate book on linear algebra. … The list of references has been enlarged considerably. The book is suitable for a second course on linear algebra and/or a graduate text, as well as a reference text.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, May, 2014)"This is the 3rd edition of a well written graduate book on linear algebra. … The book covers a wide range of topics in a moderate length and careful manner. … The list of references has been enlarged considerably. … is suitable for a second course on linear algebra and/or a graduate text, as well as a reference text." (A. Arvanitoyeorgos, Zentralblatt MATH, Vol. 1132 (10), 2008)Table of Contents* Vector Spaces * Linear Transformations * The Isomorphism Theorems * Modules I: Basic Properties * Modules II: Free and Noetherian Modules * Modules over a Principal Ideal Domain * The Structure of a Linear Operator * Eigenvalues and Eigenvectors * Real and Complex Inner Product Spaces * Structure Theory for Normal Operators * Metric Vector Spaces: The Theory of Bilinear Forms * Metric Spaces * Hilbert Spaces * Tensor Products * Positive Solutions to Linear Systems: Convexity and Separation * Affine Geometry * Operator Factorizations: QR and Singular Value * The Umbral Calculus * References * Index

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Book SynopsisThis is a short text in linear algebra, intended for a one-term course. He then starts with a discussion of linear equations, matrices and Gaussian elimination, and proceeds to discuss vector spaces, linear maps, scalar products, determinants, and eigenvalues.Trade ReviewSecond Edition S. Lang Introduction to Linear Algebra "Excellent! Rigorous yet straightforward, all answers included!"—Dr. J. Adam, Old Dominion UniversityTable of ContentsI Vectors.- II Matrices and Linear Equations.- III Vector Spaces.- IV Linear Mappings.- V Composition and Inverse Mappings.- VI Scalar Products and Orthogonality.- VII Determinants.- VIII Eigenvectors and Eigenvalues.- Answers to Exercises.

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Book SynopsisI Vector Spaces.- II Matrices.- III Linear Mappings.- IV Linear Maps and Matrices.- V Scalar Products and Orthogonality.- VI Determinants.- VII Symmetric, Hermitian, and Unitary Operators.- VIII Eigenvectors and Eigenvalues.- IX Polynomials and Matrices.- X Triangulation of Matrices and Linear Maps.- XI Polynomials and Primary Decomposition.- XII Convex Sets.- Appendix I Complex Numbers.- Appendix II Iwasawa Decomposition and Others.Trade Review"The present textbook is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of finite-dimensional vector spaces and proceeds to explain the structure theorems for linear maps, including eigenvectors and eigenvalues, quadratic and Hermitian forms, diagonalization of symmetric, Hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. It also includes a useful chapter on convex sets and the finite-dimensional Krein-Milman theorem. The presentation is aimed at the student who has already had some exposure to the elementary theory of matrices, determinants, and linear maps. In this third edition, many parts of the book have been rewritten and reorganized, and new exercises have been added." (S. Lajos, Mathematical Reviews) Table of Contents1. Vector Spaces; 2. Matrices; 3. Linear Mappings; 4. Linear Maps and Matrices; 5. Scalar Products and Orthogonality; 6. Determinants; 7. Symmetric, Hermitian, and Unitary Operators; 8. Eigenvectors and Eigenvalues; 9. Polynomials and Matrices; 10. Triangulation of Matrices and Linear Maps; 11. Polynomials and Primary Decomposition; 12. Convex Sets

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Book SynopsisThis versatile undergraduate-level text contains enough material for a one-year course and serves as a support text and reference. It combines formal theory and related computational techniques. Solutions to selected exercises. 1978 edition.

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Book SynopsisAn integrated approach to the theory of nonnegative matrices, emphasising connections with game theory, optimisation, mathematical programming, mathematical economics and statistics. The minimal prerequisites make this accessible to new graduate students.Trade Review"It is a great work; great by its dimensions, written with extreme love and care, concentrating the knowledge of a generation which was supreme in the history of matrix theory. It is a very illuminating and highly readable exposition of interesting topics which are of great relevance both to theory and applications." Mathematical Reviews Clippings 98hTable of ContentsPreface; 1. Perron-Frobenius theory and matrix games; 2. Doubly stochastic matrices; 3. Inequalities; 4. Conditionally positive definite matrices; 5. Topics in combinatorial theory; 6. Scaling problems and their applications; 7. Special matrices in economic models; References; Index; Author index.

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Book SynopsisThis book has its origins in the 1996 Durham Symposium on 'Galois representations in arithmetic algebraic geometry'. Included here are expositions of subjects on the interface between algebraic number theory and arithmetic algebraic geometry which have received substantial attention from many of the best known researchers in this field.Table of ContentsPreface; List of participants; Lecture programme; 1. The Eigencurve R. Coleman and B. Mazur; 2. Geometric trends in Galois module theory Boas Erez; 3. Mixed elliptic motives Alexander Goncharov; 4. On the Satake isomorphism Benedict H. Gross; 5. Open problems regarding rational points on curves and varieties B. Mazur; 6. Models of Shimura varieties in mixed characteristics Ben Moonen; 7. Euler systems and modular elliptic curves Karl Rubin; 8. Basic notions of rigid analytic geometry Peter Schneider; 9. An introduction to Kato's Euler systems A. J. Scholl; 10. La distribution d'Euler-Poincaré d'un groupe profini Jean-Pierre Serre.

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Book SynopsisMathematicians and historians of mathematics and science will find in The Chinese Roots of Linear Algebra new ways to conceptualize the intellectual development of linear algebra.Trade ReviewA pivotal work in the history of non-Western mathematics that will revolutionize people's understanding of the origins of techniques previously viewed as Western inventions. Choice 2011Table of ContentsPreface1. IntroductionOverview of This BookHistoriographic IssuesOutline of the Chapters2. PreliminariesChinese ConventionsChinese MathematicsModern Mathematical Terminology3. The Sources: Written Records of Early Chinese MathematicsPractices and Texts in Early Chinese MathematicsThe Book of ComputationThe Nine Chapters on the Mathematical Arts4. Excess and DeficitExcess and Deficit Problems in the Book of Computation"Excess and Deficit," Chapter 7 of the Nine Chapters5. Fangcheng, Chapter 8 of the Nine ChaptersThe Fangcheng ProcedureProcedure for Positive and Negative NumbersConclusions6. The Fangcheng Procedure in Modern Mathematical TermsConspectus of Fangcheng Problems in the Nine ChaptersEliminationBack SubstitutionIs the Fangcheng Procedure Integer-Preserving?Conclusions7. The Well ProblemTraditional Solutions to the Well ProblemThe Earliest Extant Record of a Determinantal CalculationThe Earliest Extant Record of a Determinantal SolutionConclusions8. Evidence of Early Determinantal SolutionsThe Classification of ProblemsFive Problems from the Nine ChaptersConclusions9. ConclusionsThe Early History of Linear AlgebraQuestions for Further ResearchMethodological IssuesSignificance and ImplicationsAppendix A: Examples of Similar ProblemsExamples from Diophantus's ArithmeticaExamples from ModernWorks on Linear AlgebraAppendix B: Chinese Mathematical TreatisesBibliographies of Chinese Mathematical TreatisesMathematical Treatises Listed in Chinese BibliographiesAppendix C: Outlines of ProofsBibliography of Primary and Secondary SourcesIndex

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