Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg ...

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".

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Prologue

This is Solution to Linear Algebra written by Friedberg, Insel, and Spence. And this file is generated during the Linear Algebra courses in Fall 2010 and Spring 2011. I was a TA in these courses. Although this file will be uploaded to the course website for students, the main purpose to write the solution is to do some exercises and find 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 first is that there must be several typoes and errors in this file. I would be very glad if someone send me an email, jlch3554@, 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 flied away and leave you alone.

The file is made by MikTex and Notepad++ while the graphs in this file is 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

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Version Info

? 2011, 7/27--First release with GNU Free Documentation License.

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Contents

1 Vector Spaces

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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 Differential Equations with Constant Coe-

ficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3 Elementary Matrix Operations and Systems of Linear Equa-

tions

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

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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

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