Elementary Linear Algebra - Mathematics Department

451 Pages·2016·2.71 MB·English

Elementary Linear Algebra
Kuttler
September 29, 2017
2
CONTENTS
1 Some Prerequisite Topics 1
1.1 Sets And Set Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Well Ordering And Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The Complex Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Polar Form Of Complex Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Roots Of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 The Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7 The Complex Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 The Fundamental Theorem Of Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Fn 13
2.1 Algebra in Fn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Geometric Meaning Of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Geometric Meaning Of Vector Addition . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Distance Between Points In Rn Length Of A Vector . . . . . . . . . . . . . . . . . . 17
2.5 Geometric Meaning Of Scalar Multiplication . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Parametric Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.8 Vectors And Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Vector Products 25
3.1 The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 The Geometric Significance Of The Dot Product . . . . . . . . . . . . . . . . . . . . 27
3.2.1 The Angle Between Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Work And Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.3 The Inner Product And Distance In Cn . . . . . . . . . . . . . . . . . . . . . 30
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 The Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 The Distributive Law For The Cross Product . . . . . . . . . . . . . . . . . . 37
3.4.2 The Box Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.3 Another Proof Of The Distributive Law . . . . . . . . . . . . . . . . . . . . . 39
3.5 The Vector Identity Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Systems Of Equations 43
4.1 Systems Of Equations, Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Systems Of Equations, Algebraic Procedures . . . . . . . . . . . . . . . . . . . . . . 45
4.2.1 Elementary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.2 Gauss Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.3 Balancing Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.4 Dimensionless Variables∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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4.3 Matlab And Row Reduced Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Matrices 65
5.1 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.1 Addition And Scalar Multiplication Of Matrices . . . . . . . . . . . . . . . . 65
5.1.2 Multiplication Of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.1.3 The ijth Entry Of A Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.4 Properties Of Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . 72
5.1.5 The Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1.6 The Identity And Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.7 Finding The Inverse Of A Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Matlab And Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Determinants 87
6.1 Basic Techniques And Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.1 Cofactors And 2×2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.2 The Determinant Of A Triangular Matrix . . . . . . . . . . . . . . . . . . . . 90
6.1.3 Properties Of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.1.4 Finding Determinants Using Row Operations . . . . . . . . . . . . . . . . . . 92
6.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2.1 A Formula For The Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2.2 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3 Matlab And Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7 The Mathematical Theory Of Determinants∗ 105
7.0.1 The Function sgn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.1 The Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1.2 Permuting Rows Or Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1.3 A Symmetric Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.1.4 The Alternating Property Of The Determinant . . . . . . . . . . . . . . . . . 109
7.1.5 Linear Combinations And Determinants . . . . . . . . . . . . . . . . . . . . . 110
7.1.6 The Determinant Of A Product. . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.1.7 Cofactor Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.1.8 Formula For The Inverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.1.9 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.1.10 Upper Triangular Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.2 The Cayley Hamilton Theorem∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8 Rank Of A Matrix 117
8.1 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.2 THE Row Reduced Echelon Form Of A Matrix . . . . . . . . . . . . . . . . . . . . . 123
8.3 The Rank Of A Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.3.1 The Definition Of Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.3.2 Finding The Row And Column Space Of A Matrix . . . . . . . . . . . . . . . 129
8.4 A Short Application To Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.5 Linear Independence And Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8.5.1 Linear Independence And Dependence . . . . . . . . . . . . . . . . . . . . . . 132
8.5.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.5.3 Basis Of A Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.5.4 Extending An Independent Set To Form A Basis . . . . . . . . . . . . . . . . 140
8.5.5 Finding The Null Space Or Kernel Of A Matrix . . . . . . . . . . . . . . . . 141
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8.5.6 Rank And Existence Of Solutions To Linear Systems . . . . . . . . . . . . . . 143
8.6 Fredholm Alternative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.6.1 Row, Column, And Determinant Rank . . . . . . . . . . . . . . . . . . . . . . 144
8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9 Linear Transformations 153
9.1 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.2 Constructing The Matrix Of A Linear Transformation . . . . . . . . . . . . . . . . . 155
9.2.1 Rotations in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.2.2 Rotations About A Particular Vector. . . . . . . . . . . . . . . . . . . . . . . 157
9.2.3 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.2.4 Matrices Which Are One To One Or Onto . . . . . . . . . . . . . . . . . . . . 160
9.2.5 The General Solution Of A Linear System . . . . . . . . . . . . . . . . . . . . 161
9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
10 A Few Factorizations 171
10.1 Definition Of An LU factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
10.2 Finding An LU Factorization By Inspection . . . . . . . . . . . . . . . . . . . . . . . 171
10.3 Using Multipliers To Find An LU Factorization . . . . . . . . . . . . . . . . . . . . . 172
10.4 Solving Systems Using An LU Factorization . . . . . . . . . . . . . . . . . . . . . . . 173
10.5 Justification For The Multiplier Method . . . . . . . . . . . . . . . . . . . . . . . . . 174
10.6 The PLU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
10.7 The QR Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
10.8 Matlab And Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
10.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
11 Linear Programming 185
11.1 Simple Geometric Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
11.2 The Simplex Tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
11.3 The Simplex Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
11.3.1 Maximums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
11.3.2 Minimums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
11.4 Finding A Basic Feasible Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
11.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
12 Spectral Theory 207
12.1 Eigenvalues And Eigenvectors Of A Matrix . . . . . . . . . . . . . . . . . . . . . . . 207
12.1.1 Definition Of Eigenvectors And Eigenvalues . . . . . . . . . . . . . . . . . . . 207
12.1.2 Finding Eigenvectors And Eigenvalues . . . . . . . . . . . . . . . . . . . . . . 208
12.1.3 A Warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
12.1.4 Triangular Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
12.1.5 Defective And Nondefective Matrices . . . . . . . . . . . . . . . . . . . . . . . 214
12.1.6 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
12.1.7 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
12.1.8 Complex Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
12.2 Some Applications Of Eigenvalues And Eigenvectors . . . . . . . . . . . . . . . . . . 225
12.2.1 Principal Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
12.2.2 Migration Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
12.2.3 Discrete Dynamical Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
12.3 The Estimation Of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
12.4 Matlab And Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
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13 Matrices And The Inner Product 243
13.1 Symmetric And Orthogonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
13.1.1 Orthogonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
13.1.2 Symmetric And Skew Symmetric Matrices . . . . . . . . . . . . . . . . . . . 245
13.1.3 Diagonalizing A Symmetric Matrix . . . . . . . . . . . . . . . . . . . . . . . . 250
13.2 Fundamental Theory And Generalizations . . . . . . . . . . . . . . . . . . . . . . . . 253
13.2.1 Block Multiplication Of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 253
13.2.2 Orthonormal Bases, Gram Schmidt Process . . . . . . . . . . . . . . . . . . . 257
13.2.3 Schur’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
13.3 Least Square Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
13.3.1 The Least Squares Regression Line . . . . . . . . . . . . . . . . . . . . . . . . 263
13.3.2 The Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
13.4 The Right Polar Factorization∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
13.5 The Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
13.6 Approximation In The Frobenius Norm∗ . . . . . . . . . . . . . . . . . . . . . . . . . 270
13.7 Moore Penrose Inverse∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
13.8 Matlab And Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . 273
13.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
14 Numerical Methods For Solving Linear Systems 281
14.1 Iterative Methods For Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 281
14.1.1 The Jacobi Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
14.2 Using Matlab To Iterate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
14.2.1 The Gauss Seidel Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
14.3 The Operator Norm∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
14.4 The Condition Number∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
14.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
15 Numerical Methods For Solving The Eigenvalue Problem 293
15.1 The Power Method For Eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
15.2 The Shifted Inverse Power Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
15.3 Automation With Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
15.4 The Rayleigh Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
15.5 The QR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
15.5.1 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
15.6 Matlab And The QR Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
15.6.1 The Upper Hessenberg Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
15.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
16 Vector Spaces 315
16.1 Algebraic Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
16.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
16.3 Linear Independence And Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
16.4 Vector Spaces And Fields∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
16.4.1 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
16.4.2 Polynomials And Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
16.4.3 The Algebraic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
16.4.4 The Lindemannn Weierstrass Theorem And Vector Spaces. . . . . . . . . . . 335
16.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
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17 Inner Product Spaces 341
17.1 Basic Definitions And Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
17.1.1 The Cauchy Schwarz Inequality And Norms . . . . . . . . . . . . . . . . . . . 342
17.2 The Gram Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
17.3 Approximation And Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
17.4 Orthogonal Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
17.5 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
17.6 The Discreet Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
17.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
18 Linear Transformations 359
18.1 Matrix Multiplication As A Linear Transformation . . . . . . . . . . . . . . . . . . . 359
18.2 L(V,W) As A Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
18.3 Eigenvalues And Eigenvectors Of Linear Transformations . . . . . . . . . . . . . . . 361
18.4 Block Diagonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
18.5 The Matrix Of A Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 369
18.5.1 Some Geometrically Defined Linear Transformations . . . . . . . . . . . . . . 377
18.5.2 Rotations About A Given Vector . . . . . . . . . . . . . . . . . . . . . . . . . 377
18.6 The Matrix Exponential, Differential Equations ∗ . . . . . . . . . . . . . . . . . . . . 379
18.6.1 Computing A Fundamental Matrix . . . . . . . . . . . . . . . . . . . . . . . . 385
18.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
A The Jordan Canonical Form* 395
B Directions For Computer Algebra Systems 403
B.1 Finding Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
B.2 Finding Row Reduced Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
B.3 Finding PLU Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
B.4 Finding QR Factorizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
B.5 Finding Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 403
B.6 Use Of Matrix Calculator On Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
C Answers To Selected Exercises 407
C.1 Exercises 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
C.2 Exercises 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
C.3 Exercises 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
C.4 Exercises 41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
C.5 Exercises 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
C.6 Exercises 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
C.7 Exercises 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
C.8 Exercises 147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
C.9 Exercises 164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
C.10Exercises 182 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
C.11Exercises 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
C.12Exercises 235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
C.13Exercises 273 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
C.14Exercises 289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
C.15Exercises 311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
C.16Exercises 317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
C.17Exercises 336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
C.18Exercises 353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
C.19Exercises 387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
CONTENTS
8
Preface
This is an introduction to linear algebra. The main part of the book features row operations and
everythingisdoneintermsoftherowreducedechelonformandspecificalgorithms. Attheend,the
more abstract notions of vector spaces and linear transformations on vector spaces are presented.
However, this is intended to be a first course in linear algebra for students who are sophomores
or juniors who have had a course in one variable calculus and a reasonable background in college
algebra. I have given complete proofs of all the fundamental ideas, but some topics such as Markov
matrices are not complete in this book but receive a plausible introduction. The book contains a
complete treatment of determinants and a simple proof of the Cayley Hamilton theorem although
these are optional topics. The Jordan form is presented as an appendix. I see this theorem as the
beginningofmoreadvancedtopicsinlinearalgebraandnotreallypartofabeginninglinearalgebra
course. There are extensions of many of the topics of this book in my on line book [13]. I have also
not emphasized that linear algebra can be carried out with any field although there is an optional
section on this topic, most of the book being devoted to either the real numbers or the complex
numbers. It seems to me this is a reasonable specialization for a first course in linear algebra.
Linear algebra is a wonderful interesting subject. It is a shame when it degenerates into nothing
more than a challenge to do the arithmetic correctly. It seems to me that the use of a computer
algebra system can be a great help in avoiding this sort of tedium. I don’t want to over emphasize
the use of technology, which is easy to do if you are not careful, but there are certain standard
things which are best done by the computer. Some of these include the row reduced echelon form,
PLU factorization, and QR factorization. It is much more fun to let the machine do the tedious
calculationsthantosufferwiththemyourself. However,itisnotgoodwhentheuseofthecomputer
algebra system degenerates into simply asking it for the answer without understanding what the
oracular software is doing. With this in mind, there are a few interactive links which explain how
to use a computer algebra system to accomplish some of these more tedious standard tasks. These
are obtained by clicking on the symbol I. I have included how to do it using maple and scientific
notebook because these are the two systems I am familiar with and have on my computer. Also, I
have included the very easy to use matrix calculator which is available on the web and have given
directions for Matlab at the end of relevant chapters. Other systems could be featured as well.
It is expected that people will use such computer algebra systems to do the exercises in this book
wheneveritwouldbehelpfultodoso,ratherthanwastinghugeamountsoftimedoingcomputations
by hand. However, this is not a book on numerical analysis so no effort is made to consider many
important numerical analysis issues.
I appreciate those who have found errors and needed corrections over the years that this has
been available.
There is a pdf file of this book on my web page http://www.math.byu.edu/klkuttle/ along with
some other materials soon to include another set of exercises, and a more advanced linear algebra
book. This book, as well as the more advanced text, is also available as an electronic version at
http://www.saylor.org/archivedcourses/ma211/ where it is used as an open access textbook. In
addition, it is available for free at BookBoon under their linear algebra offerings.
Elementary Linear Algebra ⃝c2012 by Kenneth Kuttler, used under a Creative Commons Attri-
bution(CCBY)licensemadepossiblebyfundingTheSaylorFoundation’sOpenTextbookChallenge
in order to be incorporated into Saylor.org’s collection of open courses available at
http://www.Saylor.org. Full license terms may be viewed at:
http://creativecommons.org/licenses/by/3.0/.
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CONTENTS
ii