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Solutions to Linear Algebra, Fourth Edition

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Solutions to Linear Algebra, Fourth Edition,Stephen H. Friedberg, Arnold J. Insel,Lawrence E. Spence
Jephian Lin, Shia Su, Zazastone LaiJuly 27, 2011
Copyright
©
2011 Chin-Hung Lin. Permission is granted to copy,distribute and/or modify this document under the terms of the GNUFree Documentation License, Version 1.3 or any later version pub-lished by the Free Software Foundation; with no Invariant Sections,no Front-Cover Texts, and no Back-Cover Texts. A copy of the li-cense is included in the section entitled “GNU Free DocumentationLicense”.1
Prologue
This is Solution to Linear Algebra written by Friedberg, Insel, and Spence. Andthis ﬁle is generated during the Linear Algebra courses in Fall 2010 and Spring2011. I was a TA in these courses. Although this ﬁle will be uploaded to thecourse website for students, the main purpose to write the solution is to do someexercises and ﬁnd some ideas about my master thesis, which is related to sometopic in graph theory called the
minimum rank problem
.Here are some important things for students and other users. The ﬁrst isthat there must be several typoes and errors in this ﬁle. I would be very glad if someone send me an email, jlch3554@hotmail.com, and give me some commentsor corrections. Second, for students, the answers here could not be the answeron any of your answer sheet of any test. The reason is the answers here aresimplistic and have some error sometimes. So it will not be a good excuse thatyour answers is as same as answers here when your scores ﬂied away and leaveyou alone.The ﬁle is made by MikTex and Notepad++ while the graphs in this ﬁleis drawn by IPE. Some answers is mostly computed by wxMaxima and littlecomputed by WolframAlpha. The English vocabulary is taught by Google Dic-tionary. I appreciate those persons who ever gave me a hand including peoplerelated to those mentioned softwares, those persons who buttressed me, and of course those instructors who ever taught me.Thanks.-Jephian LinDepartment of Mathematrics, National Taiwan University2011, 5/1
A successful and happy life requires life long hard working.
Prof. Peter Shiue2
Version Info
•
2011, 7/27—First release with GNU Free Documentation License.3
Contents
1 Vector Spaces 6
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Linear Combinations and Systems of Linear Equations . . . . . . 131.5 Linear Dependence and Linear Independence . . . . . . . . . . . . 151.6 Bases and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 171.7 Maximal Linearly Independent Subsets . . . . . . . . . . . . . . . . 24
2 Linear Transformations and Matrices 26
2.1 Linear Transformations, Null Spaces, and Ranges . . . . . . . . . 262.2 The Matrix Representation of a Linear Transformation . . . . . . 322.3 Composition of Linear Transformations and Matrix Multiplication 362.4 Invertibility and Isomorphisms . . . . . . . . . . . . . . . . . . . . . 422.5 The Change of Coordinate Matrix . . . . . . . . . . . . . . . . . . 472.6 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.7 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 . . . . . 653.2 The Rank of a Matrix and Matrix Inverses . . . . . . . . . . . . . 693.3 Systems of Linear Equation—Theoretical Aspects . . . . . . . . . 763.4 Systems of Linear Equations—Computational Aspects . . . . . . 79
4 Determinants 86
4.1 Determinants of Order 2 . . . . . . . . . . . . . . . . . . . . . . . . 864.2 Determinants of Order
n
. . . . . . . . . . . . . . . . . . . . . . . . 894.3 Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . 924.4 Summary—Important Facts about Determinants . . . . . . . . . . 1004.5 A Characterization of the Determinant . . . . . . . . . . . . . . . . 1024

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