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Unformatted text preview: Friedberg, Insel, and Spence Linear algebra, 4th ed. SOLUTIONS REFERENCE Michael L. Baker < mbaker@lambertw.com > UNIVERSITY OF WATERLOO September 11, 2010 Preface The aim of this document is to serve as a reference of problems and solutions from the fourth edition of Linear Algebra by Friedberg, Insel and Spence. Originally, I had intended the document to be used only by a student who was wellacquainted with linear algebra. However, as the document evolved, I found myself including an increasing number of problems. Therefore, I believe the document should be quite comprehensive once it is complete. I do these problems because I am interested in mathematics and consider this kind of thing to be fun. I give no guarantee that any of my solutions are the best way to approach the corresponding problems. If you find any errors (regardless of subtlety) in the document, or you have different or more elegant ways to approach something, then I urge you to contact me at the email address supplied above. This document was started on July 4, 2010. By the end of August, I expect to have covered up to the end of Chapter 5, which corresponds to the end of MATH 146, Linear Algebra 1 (Advanced Level) at the University of Waterloo. This document is currently a work in progress. 1 Contents 1 Vector Spaces 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Linear Combinations and Systems of Linear Equations . . . . . . . . . . . . . . . . . 10 1.5 Linear Dependence and Linear Independence . . . . . . . . . . . . . . . . . . . . . . 16 1.6 Bases and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.7 Maximal Linearly Independent Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Linear Transformations and Matrices 30 2.1 Linear Transformations, Null spaces, and Ranges . . . . . . . . . . . . . . . . . . . . 30 2.2 The Matrix Representation of a Linear Transformation . . . . . . . . . . . . . . . . . 34 2.3 Composition of Linear Transformations and Matrix Multiplication . . . . . . . . . . 36 2.4 Invertibility and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Appendix 37 2 1 Vector Spaces 1.1 Introduction Section 1.1 consists of an introductory, geometrically intuitive treatment of vectors (more specifi cally, Euclidean vectors). The solutions to the exercises from this section are very basic and as such have not been included in this document....
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 Fall '08
 GUREVITCH
 Linear Algebra, Algebra

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