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Linear Algebra

David Cherney, Tom Denton,

Rohit Thomas and Andrew Waldron

2

Edited by Katrina Glaeser and Travis Scrimshaw

First Edition. Davis California, 2013.This work is licensed under a

Creative Commons Attribution-NonCommercial-

ShareAlike 3.0 Unported License.

2

Contents

1 What is Linear Algebra?

9

1.1 Organizing Information

9

1.2 What are Vectors?

12

1.3 What are Linear Functions?

15

1.4 So, What is a Matrix?

20

1.4.1 Matrix Multiplication is Composition of Functions

25

1.4.2 The Matrix Detour

26

1.5 Review Problems

30

2 Systems of Linear Equations

37

2.1 Gaussian Elimination

37

2.1.1 Augmented Matrix Notation

37

2.1.2 Equivalence and the Act of Solving

40

2.1.3 Reduced Row Echelon Form

40

2.1.4 Solution Sets and RREF

45

2.2 Review Problems

48

2.3 Elementary Row Operations

52

2.3.1 EROs and Matrices

52

2.3.2 Recording EROs in (MjI). . . . . . . . . . . . . . . . 54

2.3.3 The Three Elementary Matrices

56

2.3.4LU,LDU, andPLDUFactorizations. . . . . . . . . . 58

2.4 Review Problems

61
3 4

2.5 Solution Sets for Systems of Linear Equations

63

2.5.1 The Geometry of Solution Sets: Hyperplanes

64

2.5.2 Particular Solution+Homogeneous Solutions

65

2.5.3 Solutions and Linearity

66

2.6 Review Problems

68

3 The Simplex Method

71

3.1 Pablo's Problem

71

3.2 Graphical Solutions

73

3.3 Dantzig's Algorithm

75

3.4 Pablo Meets Dantzig

78

3.5 Review Problems

80

4 Vectors in Space,n-Vectors83

4.1 Addition and Scalar Multiplication inRn. . . . . . . . . . . .84

4.2 Hyperplanes

85

4.3 Directions and Magnitudes

88

4.4 Vectors, Lists and Functions:RS. . . . . . . . . . . . . . . .94

4.5 Review Problems

97

5 Vector Spaces

101

5.1 Examples of Vector Spaces

102

5.1.1 Non-Examples

106

5.2 Other Fields

107

5.3 Review Problems

109

6 Linear Transformations

111

6.1 The Consequence of Linearity

112

6.2 Linear Functions on Hyperplanes

114

6.3 Linear Dierential Operators

115

6.4 Bases (Take 1)

115

6.5 Review Problems

118

7 Matrices

121

7.1 Linear Transformations and Matrices

121

7.1.1 Basis Notation

121

7.1.2 From Linear Operators to Matrices

127

7.2 Review Problems

129
4 5

7.3 Properties of Matrices

133

7.3.1 Associativity and Non-Commutativity

140

7.3.2 Block Matrices

142

7.3.3 The Algebra of Square Matrices

143

7.3.4 Trace

145

7.4 Review Problems

146

7.5 Inverse Matrix

150

7.5.1 Three Properties of the Inverse

150

7.5.2 Finding Inverses (Redux)

151

7.5.3 Linear Systems and Inverses

153

7.5.4 Homogeneous Systems

154

7.5.5 Bit Matrices

154

7.6 Review Problems

155

7.7 LU Redux

159

7.7.1 UsingLUDecomposition to Solve Linear Systems. . . 160

7.7.2 Finding anLUDecomposition.. . . . . . . . . . . . . 162

7.7.3 BlockLDUDecomposition. . . . . . . . . . . . . . . . 165

7.8 Review Problems

166

8 Determinants

169

8.1 The Determinant Formula

169

8.1.1 Simple Examples

169

8.1.2 Permutations

170

8.2 Elementary Matrices and Determinants

174

8.2.1 Row Swap

175

8.2.2 Row Multiplication

176

8.2.3 Row Addition

177

8.2.4 Determinant of Products

179

8.3 Review Problems

182

8.4 Properties of the Determinant

186

8.4.1 Determinant of the Inverse

190

8.4.2 Adjoint of a Matrix

190

8.4.3 Application: Volume of a Parallelepiped

192

8.5 Review Problems

193

9 Subspaces and Spanning Sets

195

9.1 Subspaces

195

9.2 Building Subspaces

197
5 6

9.3 Review Problems

202

10 Linear Independence

203

10.1 Showing Linear Dependence

204

10.2 Showing Linear Independence

207

10.3 From Dependent Independent

209

10.4 Review Problems

210

11 Basis and Dimension

213

11.1 Bases inRn.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

11.2 Matrix of a Linear Transformation (Redux)

218

11.3 Review Problems

221

12 Eigenvalues and Eigenvectors

225

12.1 Invariant Directions

227

12.2 The Eigenvalue{Eigenvector Equation

233

12.3 Eigenspaces

236

12.4 Review Problems

238

13 Diagonalization

241

13.1 Diagonalizability

241

13.2 Change of Basis

242

13.3 Changing to a Basis of Eigenvectors

246

13.4 Review Problems

248

14 Orthonormal Bases and Complements

253

14.1 Properties of the Standard Basis

253

14.2 Orthogonal and Orthonormal Bases

255

14.2.1 Orthonormal Bases and Dot Products

256

14.3 Relating Orthonormal Bases

258

14.4 Gram-Schmidt & Orthogonal Complements

261

14.4.1 The Gram-Schmidt Procedure

2 64

14.5QRDecomposition. . . . . . . . . . . . . . . . . . . . . . . . 265

14.6 Orthogonal Complements

267

14.7 Review Problems

272

15 Diagonalizing Symmetric Matrices

277

15.1 Review Problems

281
6 7

16 Kernel, Range, Nullity, Rank

285

16.1 Range

286

16.2 Image

287

16.2.1 One-to-one and Onto

289

16.2.2 Kernel

292

16.3 Summary

297

16.4 Review Problems

299

17 Least squares and Singular Values

303

17.1 Projection Matrices

306

17.2 Singular Value Decomposition

308

17.3 Review Problems

312

A List of Symbols

315

B Fields

317

C Online Resources

319

D Sample First Midterm

321

E Sample Second Midterm

331

F Sample Final Exam

341

G Movie Scripts

367

G.1 What is Linear Algebra?

367

G.2 Systems of Linear Equations

367
G.3 Vectors in Spacen-Vectors. . . . . . . . . . . . . . . . . . . . 377

G.4 Vector Spaces

379

G.5 Linear Transformations

383

G.6 Matrices

385

G.7 Determinants

395

G.8 Subspaces and Spanning Sets

403

G.9 Linear Independence

404

G.10 Basis and Dimension

407

G.11 Eigenvalues and Eigenvectors

409

G.12 Diagonalization

415

G.13 Orthonormal Bases and Complements

421
7 8

G.14 Diagonalizing Symmetric Matrices

428

G.15 Kernel, Range, Nullity, Rank

430

G.16 Least Squares and Singular Values

432
Index 432
8 1

What is Linear Algebra?

Many dicult problems can be handled easily once relevant information is organized in a certain way. This text aims to teach you how to organize in- formation in cases where certain mathematical structures are present. Linear

algebra is, in general, the study of those structures. NamelyLinear algebra is the study of vectors and linear functions.

In broad terms, vectors are things you can add and linear functions are functions of vectors that respect vector addition. The goal of this text is to teach you to organize information about vector spaces in a way that makes problems involving linear functions of many variables easy. (Or at least tractable.) To get a feel for the general idea of organizing information, of vectors, and of linear functions this chapter has brief sections on each. We start here in hopes of putting students in the right mindset for the odyssey that follows; the latter chapters cover the same material at a slower pace. Pleasequotesdbs_dbs1.pdfusesText_1

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