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Linear Algebra in Twenty Five Lectures

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Linear Algebra in Twenty Five Lectures Linear Algebra in Twenty Five Lectures Tom Denton and Andrew Waldron March 27, 2012 Edited by Katrina Glaeser, Rohit Thomas & Travis Scrimshaw 1 Contents 1 What is Linear Algebra? 12 2 Gaussian Elimination 19 2.1 Notation for Linear Systems................... 19 2.2 Reduced Row Echelon Form................... 21 3 Elementary Row Operations 27 4 Solution Sets for Systems of Linear Equations 34 4.1 Non-Leading Variables...................... 35 5 Vectors in Space, n-Vectors 43 5.1 Directions and Magnitudes.................... 46 6 Vector Spaces 53 7 Linear Transformations 58 8 Matrices 63 9 Properties of Matrices 72 9.1 Block Matrices.......................... 72 9.2 The Algebra of Square Matrices................ 73 10 Inverse Matrix 79 10.1 Three Properties of the Inverse................. 80 10.2 Finding Inverses.......................... 81 10.3 Linear Systems and Inverses................... 82 10.4 Homogeneous Systems...................... 83 10.5 Bit Matrices............................ 84 11 LU Decomposition 88 11.1 Using LU Decomposition to Solve Linear Systems....... 89 11.2 Finding an LU Decomposition.................. 90 11.3 Block LDU Decomposition.................... 94 2 12 Elementary Matrices and Determinants 96 12.1 Permutations........................... 97 12.2 Elementary Matrices....................... 100 13 Elementary Matrices and Determinants II 107 14 Properties of the Determinant 116 14.1 Determinant of the Inverse.................... 119 14.2 Adjoint of a Matrix........................ 120 14.3 Application: Volume of a Parallelepiped............ 122 15 Subspaces and Spanning Sets 124 15.1 Subspaces............................. 124 15.2 Building Subspaces........................ 126 16 Linear Independence 131 17 Basis and Dimension 139 17.1 Bases in Rn............................. 142 18 Eigenvalues and Eigenvectors 147 18.1 Matrix of a Linear Transformation............... 147 18.2 Invariant Directions........................ 151 19 Eigenvalues and Eigenvectors II 159 19.1 Eigenspaces............................ 162 20 Diagonalization 165 20.1 Diagonalization.......................... 165 20.2 Change of Basis.......................... 166 21 Orthonormal Bases 173 21.1 Relating Orthonormal Bases................... 176 22 Gram-Schmidt and Orthogonal Complements 181 22.1 Orthogonal Complements.................... 185 23 Diagonalizing Symmetric Matrices 191 3 24 Kernel, Range, Nullity, Rank 197 24.1 Summary............................. 201 25 Least Squares 206 A Sample Midterm I Problems and Solutions 211 B Sample Midterm II Problems and Solutions 221 C Sample Final Problems and Solutions 231 D Points Vs. Vectors 256 E Abstract Concepts 258 E.1 Dual Spaces............................ 258 E.2 Groups............................... 258 E.3 Fields............................... 259 E.4 Rings................................ 260 E.5 Algebras.............................. 261 F Sine and Cosine as an Orthonormal Basis 262 G Movie Scripts 264 G.1 Introductory Video........................ 264 G.2 What is Linear Algebra: Overview............... 265 G.3 What is Linear Algebra: 3 × 3 Matrix Example........ 267 G.4 What is Linear Algebra: Hint.................. 268 G.5 Gaussian Elimination: Augmented Matrix Notation...... 269 G.6 Gaussian Elimination: Equivalence of Augmented Matrices.. 270 G.7 Gaussian Elimination: Hints for Review Questions4 and5.. 271 G.8 Gaussian Elimination: 3 × 3 Example.............. 273 G.9 Elementary Row Operations: Example............. 274 G.10 Elementary Row Operations: Worked Examples........ 277 G.11 Elementary Row Operations: Explanation of Proof for Theo- rem 3.1............................... 279 G.12 Elementary Row Operations: Hint for Review Question 3.. 281 G.13 Solution Sets for Systems of Linear Equations: Planes..... 282 G.14 Solution Sets for Systems of Linear Equations: Pictures and Explanation............................ 283 4 G.15 Solution Sets for Systems of Linear Equations: Example... 285 G.16 Solution Sets for Systems of Linear Equations: Hint...... 287 G.17 Vectors in Space, n-Vectors: Overview............. 288 G.18 Vectors in Space, n-Vectors: Review of Parametric Notation. 289 G.19 Vectors in Space, n-Vectors: The Story of Your Life...... 291 G.20 Vector Spaces: Examples of Each Rule............. 292 G.21 Vector Spaces: Example of a Vector Space........... 296 G.22 Vector Spaces: Hint........................ 297 G.23 Linear Transformations: A Linear and A Non-Linear Example 298 G.24 Linear Transformations: Derivative and Integral of (Real) Poly- nomials of Degree at Most 3................... 300 G.25 Linear Transformations: Linear Transformations Hint..... 302 G.26 Matrices: Adjacency Matrix Example.............. 304 G.27 Matrices: Do Matrices Commute?................ 306 G.28 Matrices: Hint for Review Question 4.............. 307 G.29 Matrices: Hint for Review Question 5.............. 308 G.30 Properties of Matrices: Matrix Exponential Example..... 309 G.31 Properties of Matrices: Explanation of the Proof........ 310 G.32 Properties of Matrices: A Closer Look at the Trace Function. 312 G.33 Properties of Matrices: Matrix Exponent Hint......... 313 G.34 Inverse Matrix: A 2 × 2 Example................ 315 G.35 Inverse Matrix: Hints for Problem 3............... 316 G.36 Inverse Matrix: Left and Right Inverses............. 317 G.37 LU Decomposition: Example: How to Use LU Decomposition 319 G.38 LU Decomposition: Worked Example.............. 321 G.39 LU Decomposition: Block LDU Explanation.......... 322 G.40 Elementary Matrices and Determinants: Permutations.... 323 G.41 Elementary Matrices and Determinants: Some Ideas Explained 324 G.42 Elementary Matrices and Determinants: Hints for Problem 4. 327 G.43 Elementary Matrices and Determinants II: Elementary Deter- minants.............................. 328 G.44 Elementary Matrices and Determinants II: Determinants and Inverses.............................. 330 G.45 Elementary Matrices and Determinants II: Product of Deter- minants.............................. 332 G.46 Properties of the Determinant: Practice taking Determinants 333 G.47 Properties of the Determinant: The Adjoint Matrix...... 335 G.48 Properties of the Determinant: Hint for Problem 3...... 338 5 G.49 Subspaces and Spanning Sets: Worked Example........ 339 G.50 Subspaces and Spanning Sets: Hint for Problem2....... 340 G.51 Subspaces and Spanning Sets: Hint............... 342 G.52 Linear Independence: Worked Example............. 343 G.53 Linear Independence: Proof of Theorem 16.1.......... 345 G.54 Linear Independence: Hint for Problem 1............ 346 G.55 Basis and Dimension: Proof of Theorem............ 347 G.56 Basis and Dimension: Worked Example............. 348 G.57 Basis and Dimension: Hint for Problem 2............ 351 G.58 Eigenvalues and Eigenvectors: Worked Example........ 352 G.59 Eigenvalues and Eigenvectors: 2 × 2 Example......... 354 G.60 Eigenvalues and Eigenvectors: Jordan Cells........... 356 G.61 Eigenvalues and Eigenvectors II: Eigenvalues.......... 358 G.62 Eigenvalues and Eigenvectors II: Eigenspaces.......... 360 G.63 Eigenvalues and Eigenvectors II: Hint.............. 361 G.64 Diagonalization: Derivative Is Not Diagonalizable....... 362 G.65 Diagonalization: Change of Basis Example........... 363 G.66 Diagonalization: Diagionalizing Example............ 368 G.67 Orthonormal Bases: Sine and Cosine Form All Orthonormal Bases for R2 ............................ 370 G.68 Orthonormal Bases: Hint for Question2, Lecture 21...... 371 G.69 Orthonormal Bases: Hint.................... 372 G.70 Gram-Schmidt and Orthogonal Complements: 4 × 4 Gram Schmidt Example......................... 374 G.71 Gram-Schmidt and Orthogonal Complements: Overview... 376 G.72 Gram-Schmidt and Orthogonal Complements: QR Decompo- sition Example.......................... 378 G.73 Gram-Schmidt and Orthogonal Complements: Hint for Prob- lem1................................ 380 G.74 Diagonalizing Symmetric Matrices: 3 × 3 Example....... 382 G.75 Diagonalizing Symmetric Matrices: Hints for Problem 1.... 384 G.76 Kernel, Range, Nullity, Rank: Invertibility Conditions..... 385 G.77 Kernel, Range, Nullity, Rank: Hint for1........... 386 G.78 Least Squares: Hint for Problem 1............... 387 G.79 Least Squares: Hint for Problem 2............... 388 H Student Contributions 389 6 I Other Resources 390 J List of Symbols 392 Index 393 7 Preface These linear algebra lecture notes are designed to be presented as twenty five, fifty minute lectures suitable for sophomores likely to use the material for applications but still requiring a solid foundation in this fundamental branch of mathematics. The main idea of the course is to emphasize the concepts of vector spaces and linear transformations as mathematical structures that can be used to model the world around us. Once \persuaded" of this truth, students learn explicit skills such as Gaussian elimination and diagonalization in order that vectors and linear transformations become calculational tools, rather than abstract mathematics. In practical terms, the course aims to produce students who can perform computations with large linear systems while at the same time understand the concepts behind these techniques. Often-times when a problem can be re- duced to one of linear algebra it is \solved". These notes do not devote much space to applications (there are already a plethora of textbooks with titles involving some permutation of the words \linear", \algebra" and \applica- tions"). Instead, they attempt to explain the fundamental concepts
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