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Linear Algebra in Twenty Five LecturesTom Denton and Andrew Waldron March 27, 2012

Edited by Katrina Glaeser, Rohit Thomas & Travis Scrimshaw

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Contents1 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 7 Linear Transformations 8 Matrices 53 58 63

9 Properties of Matrices 72 9.1 Block Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 72 9.2 The Algebra of Square Matrices . . . . . . . . . . . . . . . . 73 10 Inverse Matrix 10.1 Three Properties of the Inverse 10.2 Finding Inverses . . . . . . . . . 10.3 Linear Systems and Inverses . . 10.4 Homogeneous Systems . . . . . 10.5 Bit Matrices . . . . . . . . . . . 79 80 81 82 83 84

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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

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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 n 17.1 Bases in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

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24 Kernel, Range, Nullity, Rank 197 24.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 25 Least Squares A Sample Midterm I Problems and Solutions B Sample Midterm II Problems and Solutions C Sample Final Problems and Solutions D Points Vs. Vectors E Abstract Concepts E.1 Dual Spaces . . E.2 Groups . . . . . E.3 Fields . . . . . E.4 Rings . . . . . . E.5 Algebras . . . . 206 211 221 231 256 258 258 258 259 260 261 262 264 . 264 . 265 . 267 . 268 . 269 . 270 . 271 . 273 . 274 . 277 . 279 . 281 . 282 . 283

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F Sine and Cosine as an Orthonormal Basis G Movie Scripts G.1 Introductory Video . . . . . . . . . . . . . . . . . . . . . . . G.2 What is Linear Algebra: Overview . . . . . . . . . . . . . . G.3 What is Linear Algebra: 3 3 Matrix Example . . . . . . . G.4 What is Linear Algebra: Hint . . . . . . . . . . . . . . . . . G.5 Gaussian Elimination: Augmented Matrix Notation . . . . . G.6 Gaussian Elimination: Equivalence of Augmented Matrices . G.7 Gaussian Elimination: Hints for Review Questions 4 and 5 . G.8 Gaussian Elimination: 3 3 Example . . . . . . . . . . . . . G.9 Elementary Row Operations: Example . . . . . . . . . . . . G.10 Elementary Row Operations: Worked Examples . . . . . . . G.11 Elementary Row Operations: Explanation of Proof for Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.12 Elementary Row Operations: Hint for Review Question 3 . G.13 Solution Sets for Systems of Linear Equations: Planes . . . . G.14 Solution Sets for Systems of Linear Equations: Pictures and Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) Polynomials 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 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 G.44 Elementary Matrices and Determinants II: Determinants and Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 G.45 Elementary Matrices and Determinants II: Product of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . G.50 Subspaces and Spanning Sets: Hint for Problem 2 . . . . . . G.51 Subspaces and Spanning Sets: Hint . . . . . . . . . . . . . . G.52 Linear Independence: Worked Example . . . . . . . . . . . . G.53 Linear Independence: Proof of Theorem 16.1 . . . . . . . . . G.54 Linear Independence: Hint for Problem 1 . . . . . . . . . . . G.55 Basis and Dimension: Proof of Theorem . . . . . . . . . . . G.56 Basis and Dimension: Worked Example . . . . . . . . . . . . G.57 Basis and Dimension: Hint for Problem 2 . . . . . . . . . . . G.58 Eigenvalues and Eigenvectors: Worked Example . . . . . . . G.59 Eigenvalues and Eigenvectors: 2 2 Example . . . . . . . . G.60 Eigenvalues and Eigenvectors: Jordan Cells . . . . . . . . . . G.61 Eigenvalues and Eigenvectors II: Eigenvalues . . . . . . . . . G.62 Eigenvalues and Eigenvectors II: Eigenspaces . . . . . . . . . G.63 Eigenvalues and Eigenvectors II: Hint . . . . . . . . . . . . . G.64 Diagonalization: Derivative Is Not Diagonalizable . . . . . . G.65 Diagonalization: Change of Basis Example . . . . . . . . . . G.66 Diagonalization: Diagionalizing Example . . . . . . . . . . . G.67 Orthonormal Bases: Sine and Cosine Form All Orthonormal Bases for R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . G.68 Orthonormal Bases: Hint for Question 2, Lecture 21 . . . . . G.69 Orthonormal Bases: Hint . . . . . . . . . . . . . . . . . . . G.70 Gram-Schmidt and Orthogonal Complements: 4 4 Gram Schmidt Example . . . . . . . . . . . . . . . . . . . . . . . . G.71 Gram-Schmidt and Orthogonal Complements: Overview . . G.72 Gram-Schmidt and Orthogonal Complements: QR Decomposition Example . . . . . . . . . . . . . . . . . . . . . . . . . G.73 Gram-Schmidt and Orthogonal Complements: Hint for Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.74 Diagonalizing Symmetric Matrices: 3 3 Exa

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