Mastering Linear Algebra: An Introduction with Applications 1056

Linear algebra may well be the most accessible of all routes into higher mathematics. It requires little more than a foundation in algebra and geometry, yet it supplies powerful tools for solving problems in subjects as diverse as computer science and chemistry, business and biology, engineering and economics, and physics and statistics, to name just a few. Furthermore, linear algebra is the gateway to almost any advanced mathematics course. Calculus, abstract algebra, real analysis, topology, number theory, and many other fields make extensive use of the central concepts of linear algebra: vector spaces and linear transformations. Professor Biography......Page 3 Course Scope......Page 7 Transformations......Page 11 Vectors......Page 28 Linear Combinations......Page 33 Abstract Vector Spaces......Page 38 The Dot Product......Page 42 Properties of the Dot Product......Page 45 A Geometric Formula for the Dot Product......Page 47 The Cross Product......Page 50 Describing Lines......Page 52 Describing Planes......Page 53 What Is a Matrix?......Page 56 Matrix Multiplication......Page 57 The Identity Matrix......Page 60 Other Matrix Properties......Page 61 Multivariable Functions......Page 65 Definition of a Linear Transformation......Page 67 Properties of Linear Transformations......Page 69 Matrix Multiplication Is a Linear Transformation......Page 72 Examples of Linear Transformations......Page 74 Linear Equations......Page 79 Systems of Linear Equations......Page 81 Solving Systems of Linear Equations......Page 82 Gaussian Elimination......Page 84 Getting Infinitely Many or No Solutions......Page 91 Quiz for Lectures 1–6......Page 94 Reduced Row Echelon Form......Page 96 Using the RREF to Find the Set of Solutions......Page 101 Row-Equivalent-Matrices......Page 104 The Span of a Set of Vectors......Page 106 When Is a Vector in the Span of a Set of Vectors?......Page 109 Linear Dependence of a Set of Vectors......Page 112 Linear Independence of a Set of Vectors......Page 115 The Null-Space of a Matrix......Page 119 Subspaces......Page 122 The Row Space and Column Space of a Matrix......Page 126 Geometric Interpretation of Row, Column, and Null-Spaces......Page 128 The Basis of a Subspace......Page 131 How to Find a Basis for a Column Space......Page 134 How to Find a Basis for a Row Space......Page 136 How to Find a Basis for a Null-Space......Page 137 The Rank-Nullity Theorem......Page 138 The Inverse of a Matrix......Page 141 Finding the Inverse of a 2 × 2 Matrix......Page 147 Properties of Inverses......Page 149 The Importance of Invertible Matrices......Page 151 Finding the Inverse of an n × n Matrix......Page 152 Criteria for Telling If a Matrix Is Invertible......Page 156 Quiz for Lectures 7–12......Page 158 The 1 × 1 and 2 × 2 Determinants......Page 160 The 3 × 3 Determinant......Page 162 The n × n Determinant......Page 168 Calculating Determinants Quickly......Page 169 The Geometric Meaning of the n × n Determinant......Page 171 Consequences......Page 173 Population Dynamics Application......Page 175 Understanding Matrix Powers......Page 179 Eigenvectors and Eigenvalues......Page 181 Solving the Eigenvector Equation......Page 182 Return to Population Dynamics Application......Page 183 Lecture 15—Eigenvectors and Eigenvalues: Geometry......Page 185 The Geometry of Eigenvectors and Eigenvalues......Page 186 Verifying That a Vector Is an Eigenvector......Page 188 Finding Eigenvectors and Eigenvalues......Page 189 Matrix Powers......Page 194 Change of Basis......Page 197 Eigenvalues and the Determinant......Page 200 Algebraic Multiplicity and Geometric Multiplicity......Page 202 Diagonalizability......Page 204 Similar Matrices......Page 207 Recalling the Population Dynamics Model......Page 209 High Predation......Page 214 Low Predation......Page 216 Medium Predation......Page 220 Lecture 18—Differential Equations: New Applications......Page 222 Solving a System of Differential Equations......Page 223 Complex Eigenvalues......Page 226 Quiz for Lectures 13–18......Page 230 Orthogonal Sets......Page 232 Orthogonal Matrices......Page 236 Properties of Orthogonal Matrices......Page 237 The Gram-Schmidt Process......Page 238 QR-Factorization......Page 240 Orthogonal Diagonalization......Page 241 Lecture 20—Markov Chains: Hopping Around......Page 243 Markov Chains......Page 244 Economic Mobility......Page 245 Theorems about Markov Chains......Page 247 Single-Variable Calculus......Page 252 Multivariable Functions......Page 254 Differentiability......Page 256 The Derivative......Page 257 Chain Rule......Page 259 Lecture 22—Multilinear Regression: Least Squares......Page 261 Linear Regression......Page 262 Multiple Linear Regression......Page 269 Invertibility of the Gram Matrix......Page 270 How Good Is the Fit?......Page 272 Polynomial Regression......Page 274 The Singular Value Decomposition......Page 276 The Geometric Meaning of the SVD......Page 279 Computing the SVD......Page 283 Functions as Vectors......Page 286 General Vector Spaces......Page 288 Fibonacci-Type Sequences as a Vector Space......Page 289 Space of Functions as Vector Spaces......Page 292 Solutions of Differential Equations......Page 294 Ideas of Fourier Analysis......Page 297 Quiz for Lectures 19–24......Page 300 Lectures 1–6......Page 302 Lectures 7–12......Page 304 Lectures 13–18......Page 307 Lectures 19–24......Page 309 Bibliography......Page 311 Professor Biography 3 Course Scope 7 Lecture 1—Linear Algebra: Powerful Transformations 11 Transformations 11 Lecture 2—Vectors: Describing Space and Motion 28 Vectors 28 Linear Combinations 33 Abstract Vector Spaces 38 Lecture 3—Linear Geometry: Dots and Crosses 42 The Dot Product 42 Properties of the Dot Product 45 A Geometric Formula for the Dot Product 47 The Cross Product 50 Describing Lines 52 Describing Planes 53 Lecture 4—Matrix Operations 56 What Is a Matrix? 56 Matrix Multiplication 57 The Identity Matrix 60 Other Matrix Properties 61 Lecture 5—Linear Transformations 65 Multivariable Functions 65 Definition of a Linear Transformation 67 Properties of Linear Transformations 69 Matrix Multiplication Is a Linear Transformation 72 Examples of Linear Transformations 74 Lecture 6—Systems of Linear Equations 79 Linear Equations 79 Systems of Linear Equations 81 Solving Systems of Linear Equations 82 Gaussian Elimination 84 Getting Infinitely Many or No Solutions 91 Quiz for Lectures 1–6 94 Lecture 7—Reduced Row Echelon Form 96 Reduced Row Echelon Form 96 Using the RREF to Find the Set of Solutions 101 Row-Equivalent-Matrices 104 Lecture 8—Span and Linear Dependence 106 The Span of a Set of Vectors 106 When Is a Vector in the Span of a Set of Vectors? 109 Linear Dependence of a Set of Vectors 112 Linear Independence of a Set of Vectors 115 Lecture 9—Subspaces: Special Subsets to Look For 119 The Null-Space of a Matrix 119 Subspaces 122 The Row Space and Column Space of a Matrix 126 Lecture 10—Bases: Basic Building Blocks 128 Geometric Interpretation of Row, Column, and Null-Spaces 128 The Basis of a Subspace 131 How to Find a Basis for a Column Space 134 How to Find a Basis for a Row Space 136 How to Find a Basis for a Null-Space 137 The Rank-Nullity Theorem 138 Lecture 11—Invertible Matrices: Undoing What You Did 141 The Inverse of a Matrix 141 Finding the Inverse of a 2 × 2 Matrix 147 Properties of Inverses 149 Lecture 12—The Invertible Matrix Theorem 151 The Importance of Invertible Matrices 151 Finding the Inverse of an n × n Matrix 152 Criteria for Telling If a Matrix Is Invertible 156 Quiz for Lectures 7–12 158 Lecture 13—Determinants: Numbers That Say a Lot 160 The 1 × 1 and 2 × 2 Determinants 160 The 3 × 3 Determinant 162 The n × n Determinant 168 Calculating Determinants Quickly 169 The Geometric Meaning of the n × n Determinant 171 Consequences 173 Lecture 14—Eigenstuff: Revealing Hidden Structure 175 Population Dynamics Application 175 Understanding Matrix Powers 179 Eigenvectors and Eigenvalues 181 Solving the Eigenvector Equation 182 Return to Population Dynamics Application 183 Lecture 15—Eigenvectors and Eigenvalues: Geometry 185 The Geometry of Eigenvectors and Eigenvalues 186 Verifying That a Vector Is an Eigenvector 188 Finding Eigenvectors and Eigenvalues 189 Matrix Powers 194 Lecture 16—Diagonalizability 197 Change of Basis 197 Eigenvalues and the Determinant 200 Algebraic Multiplicity and Geometric Multiplicity 202 Diagonalizability 204 Computing Matrix Powers 207 Similar Matrices 207 Lecture 17—Population Dynamics: Foxes and Rabbits 209 Recalling the Population Dynamics Model 209 High Predation 214 Low Predation 216 Medium Predation 220 Lecture 18—Differential Equations: New Applications 222 Solving a System of Differential Equations 223 Complex Eigenvalues 226 Quiz for Lectures 13–18 230 Lecture 19—Orthogonality: Squaring Things Up 232 Orthogonal Sets 232 Orthogonal Matrices 236 Properties of Orthogonal Matrices 237 The Gram-Schmidt Process 238 QR-Factorization 240 Orthogonal Diagonalization 241 Lecture 20—Markov Chains: Hopping Around 243 Markov Chains 244 Economic Mobility 245 Theorems about Markov Chains 247 Lecture 21—Multivariable Calculus: Derivative Matrix 252 Single-Variable Calculus 252 Multivariable Functions 254 Differentiability 256 The Derivative 257 Chain Rule 259 Lecture 22—Multilinear Regression: Least Squares 261 Linear Regression 262 Multiple Linear Regression 269 Invertibility of the Gram Matrix 270 How Good Is the Fit? 272 Polynomial Regression 274 Lecture 23—Singular Value Decomposition: So Cool 276 The Singular Value Decomposition 276 The Geometric Meaning of the SVD 279 Computing the SVD 283 Lecture 24—General Vector Spaces: More to Explore 286 Functions as Vectors 286 General Vector Spaces 288 Fibonacci-Type Sequences as a Vector Space 289 Space of Functions as Vector Spaces 292 Solutions of Differential Equations 294 Ideas of Fourier Analysis 297 Quiz for Lectures 19–24 300 Solutions 302 Lectures 1–6 302 Lectures 7–12 304 Lectures 13–18 307 Lectures 19–24 309 Bibliography 311