Fundamentals of Matrix Algebra, 2011a

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Contents 5 Graphical Exploraons of Vectors 187 5.1 Transformaons of the Cartesian Plane . . . . . . . . . . . . . . . . . 187 5.2 Properes of Linear Transformaons . . . . . . . . . . . . . . . . . . 202 5.3 Visualizing Vectors: Vectors in Three Dimensions . . . . . . . . . . . 215 A Soluons To Selected Problems 227 Index 237 viii
1 . S L E You have probably encountered systems of linear equaons before; you can probably remember solving systems of equaons where you had three equaons, three unknowns, and you tried to find the value of the unknowns. In this chapter we will uncover some of the fundamental principles guiding the soluon to such problems. Solving such systems was a bit me consuming, but not terribly difficult. So why bother? We bother because linear equaons have many, many, many applicaons, from business to engineering to computer graphics to understanding more mathematics. And not only are there many applicaons of systems of linear equaons, on most occasions where these systems arise we are using far more than three variables. (Engineering applicaons, for instance, oen require thousands of variables.) So geng a good understanding of how to solve these systems effecvely is important. But don’t worry; we’ll start at the beginning. 1.1 Introducon to Linear Equaons AS YOU READ ... . . . 1. What is one of the annoying habits of mathemacians? 2. What is the difference between constants and coefficients? 3. Can a coefficient in a linear equaon be 0? We’ll begin this secon by examining a problem you probably already know how to solve. . Example 1 .Suppose a jar contains red, blue and green marbles. You are told that there are a total of 30 marbles in the jar; there are twice as many red marbles as
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Chapter 3 Operaons on Matrices .
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Chapter 3 Operaons on Matrices oper
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Chapter 3 Operaons on Matrices We e
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Chapter 3 Operaons on Matrices S To
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Chapter 3 Operaons on Matrices C 1,
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Chapter 3 Operaons on Matrices ⎡
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Chapter 4 Eigenvalues and Eigenvect
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Chapter 4 . Eigenvalues and Eigenve
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5 . G E V We already looked at the
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5.1 Transformaons of the Cartesian
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5.2 Properes of Linear Transformaon
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5.3 Visualizing Vectors: Vectors in
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A . S T S P Chapter 1 Secon 1.1 1.
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[ ] 9 −7 5. 11 −6 [ ] −14 7.
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1. Mulply A⃗u and A⃗v to verify
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21. A is diagonal, as is A T . ⎡
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(d) -1 (e) 0 ⎡ 5. (a) λ 1 = −4
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Index ansymmetric, 128 augmented ma
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Fundamentals of Matrix Algebra, 2011a

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