Exam (elaborations)

SOLUTIONS MANUAL for Finite-Dimensional Linear Algebra 1st Edition by Mark Gockenbach | All 10 Chapters

29 views 0 purchase

Course
Finite-Dimensional Linear Algebra 1st Edition

Institution
Finite-Dimensional Linear Algebra 1st Edition

Book
Finite-Dimensional Linear Algebra

SOLUTIONS MANUAL for Finite-Dimensional Linear Algebra 1st Edition by Mark Gockenbach. ISBN 9781439815649, ISBN-. _ TABLE OF CONTENTS_ CHAPTERS 1: Some Problems Posed on Vector Space s CHAPTERS 2: Fields and Vector Spaces CHAPTERS 3: Linear Operators CHAPTERS 4: Determinants and Eigenvalues CHAPTER...

[Show more]

Preview 4 out of 260 pages

View example

Uploaded on September 25, 2023
Number of pages 260
Written in 2023/2024
Type Exam (elaborations)
Contains Questions & answers

isbn 9781439815649
isbn 978 1439815632
finite dimensional linear algebra 1st edition
finite dimensional linear algebra
solutions manual for finite dimensional linear
solutions manual

Institution Finite-Dimensional Linear Algebra 1st Edition
Course Finite-Dimensional Linear Algebra 1st Edition

AcademiContent Member since 5 year 2440 documents sold

167

$32.42

Added

Add to cart Add to wishlist

100% satisfaction guarantee
Immediately available after payment
Both online and in PDF
No strings attached

,SOLUTIONS MANUAL FOR
Finite Dimensional
Linear Algebra

by
Mark S. Gockenbach
Michigan Technological University

,Contents

Errata for the first printing 1

2 Fields and vector spaces 5
2.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Linear combinations and spanning sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Basis and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Properties of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8 Polynomial interpolation and the Lagrange basis . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.9 Continuous piecewise polynomial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Linear operators 47
3.1 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 More properties of linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Isomorphic vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4 Linear operator equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6 The fundamental theorem; inverse operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.7 Gaussian elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.8 Newton’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.9 Linear ordinary diﬀerential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.10 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.11 Coding theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.12 Linear programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4 Determinants and eigenvalues 91
4.1 The determinant function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Further properties of the determinant function . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Practical computation of det(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.5 Eigenvalues and the characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.6 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.7 Eigenvalues of linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.8 Systems of linear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.9 Integer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5 The Jordan canonical form 117
5.1 Invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2 Generalized eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3 Nilpotent operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4 The Jordan canonical form of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

i

, ii CONTENTS

5.5 The matrix exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.6 Graphs and eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6 Orthogonality and best approximation 149
6.1 Norms and inner products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 The adjoint of a linear operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3 Orthogonal vectors and bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.4 The projection theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.5 The Gram-Schmidt process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.6 Orthogonal complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.7 Complex inner product spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.8 More on polynomial approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.9 The energy inner product and Galerkin’s method . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.10 Gaussian quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.11 The Helmholtz decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7 The spectral theory of symmetric matrices 193
7.1 The spectral theorem for symmetric matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.2 The spectral theorem for normal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.3 Optimization and the Hessian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.4 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.5 Spectral methods for diﬀerential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8 The singular value decomposition 209
8.1 Introduction to the SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.2 The SVD for general matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.3 Solving least-squares problems using the SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.4 The SVD and linear inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.5 The Smith normal form of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

9 Matrix factorizations and numerical linear algebra 223
9.1 The LU factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9.2 Partial pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
9.3 The Cholesky factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
9.4 Matrix norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9.5 The sensitivity of linear systems to errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.6 Numerical stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
9.7 The sensitivity of the least-squares problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
9.8 The QR factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
9.9 Eigenvalues and simultaneous iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
9.10 The QR algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

10 Analysis in vector spaces 247
10.1 Analysis in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
10.2 Inﬁnite-dimensional vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
10.3 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
10.4 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller AcademiContent. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for $32.42. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

67096 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling

Popular Universities in the United States

Popular books

Find notes and summaries for these qualifications

Exam (elaborations)

SOLUTIONS MANUAL for Finite-Dimensional Linear Algebra 1st Edition by Mark Gockenbach | All 10 Chapters

Document information

Subjects

Connected book

More summaries for

Written for

Seller

Reviews received

Content preview