Fundamentals of Matrix Algebra, 2011a

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Chapter 1 Systems of Linear Equaons problems”). Exercises 1.4 In Exercises 1 – 14, find the soluon to the given linear system. If the system has infinite soluons, give 2 parcular soluons. 1. 2. 3. 4. 5. 6. 7. 8. 9. 2x 1 + 4x 2 = 2 x 1 + 2x 2 = 1 −x 1 + 5x 2 = 3 2x 1 − 10x 2 = −6 x 1 + x 2 = 3 2x 1 + x 2 = 4 −3x 1 + 7x 2 = −7 2x 1 − 8x 2 = 8 2x 1 + 3x 2 = 1 −2x 1 − 3x 2 = 1 x 1 + 2x 2 = 1 −x 1 − 2x 2 = 5 −2x 1 + 4x 2 + 4x 3 = 6 x 1 − 3x 2 + 2x 3 = 1 −x 1 + 2x 2 + 2x 3 = 2 2x 1 + 5x 2 + x 3 = 2 −x 1 − x 2 + x 3 + x 4 = 0 −2x 1 − 2x 2 + x 3 = −1 11. 12. 13. 14. 2x 1 + x 2 + 2x 3 = 0 x 1 + x 2 + 3x 3 = 1 3x 1 + 2x 2 + 5x 3 = 3 x 1 + 3x 2 + 3x 3 = 1 2x 1 − x 2 + 2x 3 = −1 4x 1 + 5x 2 + 8x 3 = 2 x 1 + 2x 2 + 2x 3 = 1 2x 1 + x 2 + 3x 3 = 1 3x 1 + 3x 2 + 5x 3 = 2 2x 1 + 4x 2 + 6x 3 = 2 1x 1 + 2x 2 + 3x 3 = 1 −3x 1 − 6x 2 − 9x 3 = −3 In Exercises 15 – 18, state for which values of k the given system will have exactly 1 soluon, infinite soluons, or no soluon. 15. 16. 17. x 1 + 2x 2 = 1 2x 1 + 4x 2 = k x 1 + 2x 2 = 1 x 1 + kx 2 = 1 x 1 + 2x 2 = 1 x 1 + kx 2 = 2 10. x 1 + x 2 + 6x 3 + 9x 4 = 0 −x 1 − x 3 − 2x 4 = −3 18. x 1 + 2x 2 = 1 x 1 + 3x 2 = k 1.5 Applicaons of Linear Systems AS YOU READ ... . . . 1. How do most problems appear “in the real world?” 2. The unknowns in a problem are also called what? 3. How many points are needed to determine the coefficients of a 5 th degree polynomial? We’ve started this chapter by addressing the issue of finding the soluon to a system of linear equaons. In subsequent secons, we defined matrices to store linear 34
1.5 Applicaons of Linear Systems equaon informaon; we described how we can manipulate matrices without changing the soluons; we described how to efficiently manipulate matrices so that a working soluon can be easily found. We shouldn’t lose sight of the fact that our work in the previous secons was aimed at finding soluons to systems of linear equaons. In this secon, we’ll learn how to apply what we’ve learned to actually solve some problems. Many, many, many problems that are addressed by engineers, businesspeople, sciensts and mathemacians can be solved by properly seng up systems of linear equaons. In this secon we highlight only a few of the wide variety of problems that matrix algebra can help us solve. We start with a simple example. . Example 16 .A jar contains 100 blue, green, red and yellow marbles. There are twice as many yellow marbles as blue; there are 10 more blue marbles than red; the sum of the red and yellow marbles is the same as the sum of the blue and green. How many marbles of each color are there? S Let’s call the number of blue balls b, and the number of the other balls g, r and y, each represenng the obvious. Since we know that we have 100 marbles, we have the equaon b + g + r + y = 100. The next sentence in our problem statement allows us to create three more equaons. We are told that there are twice as many yellow marbles as blue. One of the following two equaons is correct, based on this statement; which one is it? 2y = b or 2b = y The first equaon says that if we take the number of yellow marbles, then double it, we’ll have the number of blue marbles. That is not what we were told. The second equaon states that if we take the number of blue marbles, then double it, we’ll have the number of yellow marbles. This is what we were told. The next statement of “there are 10 more blue marbles as red” can be wrien as either b = r + 10 or r = b + 10. Which is it? The first equaon says that if we take the number of red marbles, then add 10, we’ll have the number of blue marbles. This is what we were told. The next equaon is wrong; it implies there are more red marbles than blue. The final statement tells us that the sum of the red and yellow marbles is the same as the sum of the blue and green marbles, giving us the equaon r + y = b + g. 35
Page 1: . . . Fundamentals of Matrix Algebr
Page 4 and 5: Copyright © 2011 Gregory Hartman L
Page 7: P A Note to Students, Teachers, and
Page 10 and 11: Contents 5 Graphical Exploraons of
Page 12 and 13: Chapter 1 Systems of Linear Equaons
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Chapter 2 Matrix Arithmec y ⃗v .
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Chapter 2 Matrix Arithmec Again, In
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Chapter 2 Matrix Arithmec equaon is
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Chapter 2 Matrix Arithmec . Key Ide
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Chapter 2 Matrix Arithmec “x 3
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Chapter 2 Matrix Arithmec . Example
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Chapter 2 Matrix Arithmec [ ] 1 −
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Chapter 2 Matrix Arithmec We know h
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Chapter 2 Matrix Arithmec Noce also
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Chapter 2 Matrix Arithmec from abov
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Chapter 2 Matrix Arithmec 4. T/F: I
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Chapter 2 Matrix Arithmec Since mat
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Chapter 2 Matrix Arithmec We have a
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Chapter 2 Matrix Arithmec . Ṫheor
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Chapter 2 Matrix Arithmec ⎡ −15
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Chapter 2 Matrix Arithmec is that t
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Chapter 2 Matrix Arithmec We now ap
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Chapter 2 Matrix Arithmec more to b
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Chapter 2 Matrix Arithmec can cause
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Chapter 3 Operaons on Matrices .
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Chapter 3 Operaons on Matrices Ther
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Chapter 3 Operaons on Matrices Find
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Chapter 3 Operaons on Matrices Let
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Chapter 3 Operaons on Matrices oper
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Chapter 3 Operaons on Matrices The
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Chapter 3 Operaons on Matrices We e
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Chapter 3 Operaons on Matrices In t
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Chapter 3 Operaons on Matrices S To
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Chapter 3 Operaons on Matrices We
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Chapter 3 Operaons on Matrices C 1,
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Chapter 3 Operaons on Matrices C 1,
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Chapter 3 Operaons on Matrices 21.
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Chapter 3 Operaons on Matrices S At
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Chapter 3 Operaons on Matrices .
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Chapter 3 Operaons on Matrices 1 2
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Chapter 3 Operaons on Matrices Obvi
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Chapter 3 Operaons on Matrices . Ke
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Chapter 3 Operaons on Matrices ⎡
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Chapter 3 Operaons on Matrices S Ru
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Chapter 3 Operaons on Matrices read
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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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