# la - Friedberg, Insel, and Spence Linear algebra, 4th ed....

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Friedberg, Insel, and Spence Linear algebra, 4th ed. SOLUTIONS REFERENCE Michael L. Baker < [email protected] > UNIVERSITY OF WATERLOO January 23, 2011 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 well-acquainted 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 e-mail 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

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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
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. 1.2 Vector Spaces Section 1.2 introduces an algebraic structure known as a vector space over a ﬁeld , which is then used to provide a more abstract notion of a vector (namely, as an element of such an algebraic structure). Matrices and n -tuples are also introduced. Some elementary theorems are stated and proved, such as the Cancellation Law for Vector Addition (Theorem 1.1), in addition to a few uniqueness results concerning additive identities and additive inverses. 8. In any vector space V , show that ( a + b )( x + y ) = ax + ay + bx + by for any x,y V and any a,b F . Solution . Noting that ( a + b ) F , we have ( a + b )( x + y ) = ( a + b ) x + ( a + b ) y (by VS 7) = ax + bx + ay + by (by VS 8) = ax + ay + bx + by (by VS 1) as required. 9. Prove Corollaries 1 and 2 [uniqueness of additive identities and additive inverses] of Theorem 1.1 and Theorem 1.2(c) [ a 0 = 0 for each a F ].

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## This note was uploaded on 05/20/2011 for the course MATH 115A taught by Professor Liu during the Spring '07 term at UCLA.

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la - Friedberg, Insel, and Spence Linear algebra, 4th ed....

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