Solutions to Linear Algebra - Version 1 - First release

Sep 22, 2026

Uploaded by Bulrr

Solutions to Linear Algebra (Fourth Edition) provides course-oriented worked solutions for major topics spanning vector spaces, linear transformations, matrices, systems of linear equations, determinants, diagonalization, inner product spaces, and canonical forms. The preface explains its origin from Linear Algebra courses in Fall 2010 and Spring 2011, notes that errors and typos may exist, and emphasizes that answers are not meant to replace official test solutions. The document also documents tooling used and distribution under the GNU Free Documentation License (version 1.3+).

LanguageEnglish

CategoryExam

FormatPDF

Pages276

Reading Time11 h 36 min

AccessFree to download

Prologue

Version Info

Contents

Vector Spaces

Linear Transformations and Matrices

Elementary Matrix Operations and Systems of Linear Equations

Determinants

Diagonalization

Inner Product Spaces

Canonical Forms

GNU Free Documentation License

Applicability and Definitions

Verbatim Copying

Copying in Quantity

Modifications

Combining Documents

Collections of Documents

Aggregation with Independent Works

Translation

Termination

Future Revisions of this License

Relicensing

Appendices

Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence Jephian Lin, Shia Su, Zazastone Lai July 27, 2011 Copyright © 2011 Chin-Hung Lin. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled \GNU Free Documentation License". Prologue This is Solution to Linear Algebra written by Friedberg, Insel, and Spence. And this 􀀌le is generated during the Linear Algebra courses in Fall 2010 and Spring 2011. I was a TA in these courses. Although this 􀀌le will be uploaded to the course website for students, the main purpose to write the solution is to do some exercises and 􀀌nd some ideas about my master thesis, which is related to some topic in graph theory called the minimum rank problem. Here are some important things for students and other users. The 􀀌rst is that there must be several typoes and errors in this 􀀌le. I would be very glad if someone send me an email, [jlch3554@hotmail.com](jlch3554@hotmail.com), and give me some comments or corrections. Second, for students, the answers here could not be the answer on any of your answer sheet of any test. The reason is the answers here are simplistic and have some error sometimes. So it will not be a good excuse that your answers is as same as answers here when your scores 􀀍ied away and leave you alone. The 􀀌le is made by MikTex and Notepad++ while the graphs in this 􀀌leis drawn by IPE. Some answers is mostly computed by wxMaxima and little computed by WolframAlpha. The English vocabulary is taught by Google Dictionary. I appreciate those persons who ever gave me a hand including people related to those mentioned softwares, those persons who buttressed me, and of course those instructors who ever taught me. Thanks. -Jephian Lin Department of Mathematrics, National Taiwan University 2011, 5/1 A successful and happy life requires life long hard working. Prof. Peter Shiue Version Info • 2011, 7/27|First release with GNU Free Documentation License. Contents 1 Vector Spaces 6 1.1 Introduction ................................ 6 1.2 Vector Spaces ............................... 7 1.3 Subspaces ................................. 9 1.4 Linear Combinations and Systems of Linear Equations ...... 13 1.5 Linear Dependence and Linear Independence ............ 15 1.6 Bases and Dimension .......................... 17 1.7 Maximal Linearly Independent Subsets ................ 24 2 Linear Transformations and Matrices 26 2.1 Linear Transformations, Null Spaces, and Ranges ......... 26 2.2 The Matrix Representation of a Linear Transformation ...... 32 2.3 Composition of Linear Transformations and Matrix Multiplication 36 2.4 Invertibility and Isomorphisms ..................... 42 2.5 The Change of Coordinate Matrix .................. 47 2.6 Dual Spaces ................................ 50 2.7 Homogeneous Linear Di􀀋erential Equations with Constant Coe- 􀀌cients ................................... 58 3 Elementary Matrix Operations and Systems of Linear Equations 65 3.1 Elementary Matrix Operations and Elementary Matrices ..... 65 3.2 The Rank of a Matrix and Matrix Inverses ............. 69 3.3 Systems of Linear Equation|Theoretical Aspects ......... 76 3.4 Systems of Linear Equations|Computational Aspects ...... 79 4 Determinants 86 4.1 Determinants of Order 2 ........................ 86 4.2 Determinants of Order n ........................ 89 4.3 Properties of Determinants ....................... 92 4.4 Summary|Important Facts about Determinants .......... 100 4.5 A Characterization of the Determinant ................ 102 5 Diagonalization 108 5.1 Eigenvalues and Eigenvectors ..................... 108 5.

FAQs

What is the purpose of this linear algebra solutions file?

It was written to support exercises and to gather ideas for a master’s thesis related to the minimum rank problem in graph theory.

Are the provided answers guaranteed to be correct for exams?

No. The author warns that the solutions may be simplistic and sometimes contain errors, so they should not be used as official exam answers.

Under what license is the document distributed?

The document is distributed under the GNU Free Documentation License, Version 1.3 or any later version, with stated conditions such as no Invariant Sections and no Front/Back-Cover Texts.

Sep 22, 2026

Uploaded by Bulrr