Fundamentals of Matrix Algebra, 2011a

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Chapter 1 Systems of Linear Equaons is chosen, the value of x 1 is determined. We have infinite choices for the value of x 2 , so therefore we have infinite soluons. For example, if we set x 2 = 0, then x 1 = 1; if we set x 2 = 5, then x 1 = −4. . Let’s try another example, one that uses more variables. . Example 9 Find the soluon to the linear system x 2 − x 3 = 3 x 1 + 2x 3 = 2 −3x 2 + 3x 3 = −9 . S To find the soluon, put the corresponding matrix into reduced row echelon form. ⎡ ⎤ ⎡ ⎤ ⎣ 0 1 −1 3 1 0 2 2 ⎦ −→ rref ⎣ 1 0 2 2 0 1 −1 3 ⎦ 0 −3 3 −9 0 0 0 0 Now convert this reduced matrix back into equaons. We have or, equivalently, x 1 + 2x 3 = 2 x 2 − x 3 = 3 x 1 = 2 − 2x 3 x 2 = 3 + x 3 x 3 is free. These two equaons tell us that the values of x 1 and x 2 depend on what x 3 is. As we saw before, there is no restricon on what x 3 must be; it is “free” to take on the value of any real number. Once x 3 is chosen, we have a soluon. Since we have infinite choices for the value of x 3 , we have infinite soluons. As examples, x 1 = 2, x 2 = 3, x 3 = 0 is one soluon; x 1 = −2, x 2 = 5, x 3 = 2 is another soluon. Try plugging these values back into the original equaons to verify that these indeed are soluons. (By the way, since infinite soluons exist, this system of equaons is consistent.) . In the two previous examples we have used the word “free” to describe certain variables. What exactly is a free variable? How do we recognize which variables are free and which are not? Look back to the reduced matrix in Example 8. Noce that there is only one leading 1 in that matrix, and that leading 1 corresponded to the x 1 variable. That told us that x 1 was not a free variable; since x 2 did not correspond to a leading 1, it was a free variable. 26
1.4 Existence and Uniqueness of Soluons Look also at the reduced matrix in Example 9. There were two leading 1s in that matrix; one corresponded to x 1 and the other to x 2 . This meant that x 1 and x 2 were not free variables; since there was not a leading 1 that corresponded to x 3 , it was a free variable. We formally define this and a few other terms in this following definion. . Ḋefinion 6 Dependent and Independent Variables Consider the reduced row echelon form of an augmented matrix of a linear system of equaons. Then: . a variable that corresponds to a leading 1 is a basic, or dependent, variable, and a variable that does not correspond to a leading 1 is a free, or independent, variable. One can probably see that “free” and “independent” are relavely synonymous. It follows that if a variable is not independent, it must be dependent; the word “basic” comes from connecons to other areas of mathemacs that we won’t explore here. These definions help us understand when a consistent system of linear equaons will have infinite soluons. If there are no free variables, then there is exactly one soluon; if there are any free variables, there are infinite soluons. . Consistent Soluon Types . Key Idea 2 . A consistent linear system of equaons will have exactly one soluon if and only if there is a leading 1 for each variable in the system. If a consistent linear system of equaons has a free variable, it has infinite soluons. If a consistent linear system has more variables than leading 1s, then the system will have infinite soluons. A consistent linear system with more variables than equaons will always have infinite soluons. Note: Key Idea 2 applies only to consistent systems. If a system is inconsistent, 27
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Page 4 and 5: Copyright © 2011 Gregory Hartman L
Page 7: P A Note to Students, Teachers, and
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Chapter 2 Matrix Arithmec y A⃗x A
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Chapter 2 Matrix Arithmec Of course
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Chapter 2 Matrix Arithmec 2.4 Vecto
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Chapter 2 Matrix Arithmec In previo
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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 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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