Mastering Linear Algebra: An Introduction with Applications

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