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 Questions 4 and 5 . . 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
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