Fundamentals of Matrix Algebra, 2011a

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Chapter 1 Systems of Linear Equaons row echelon form helps us idenfy all types of soluons. We’ll explore the topic of understanding what the reduced row echelon form of a matrix tells us in the following secons; in this secon we focus on finding it. . Ḋefinion 3 Reduced Row Echelon Form A matrix is in reduced row echelon form if its entries sasfy the following condions. 1. The first nonzero entry in each row is a 1 (called a leading 1). . 2. Each leading 1 comes in a column to the right of the leading 1s in rows above it. 3. All rows of all 0s come at the boom of the matrix. 4. If a column contains a leading 1, then all other entries in that column are 0. A matrix that sasfies the first three condions is said to be in row echelon form. . Example 3 .Which of the following matrices is in reduced row echelon form? [ ] [ ] 1 0 1 0 1 a) b) 0 1 0 1 2 c) [ ] 0 0 0 0 d) [ 1 1 ] 0 0 0 1 ⎡ e) ⎣ 1 0 0 1 ⎤ ⎡ 0 0 0 0 ⎦ f) ⎣ 1 2 0 0 ⎤ 0 0 3 0 ⎦ 0 0 1 3 0 0 0 4 ⎡ 0 1 2 3 0 ⎤ 4 ⎡ 1 1 ⎤ 0 g) ⎣ 0 0 0 0 1 5 ⎦ h) ⎣ 0 1 0 ⎦ 0 0 0 0 0 0 0 0 1 S The matrices in a), b), c), d) and g) are all in reduced row echelon form. Check to see that each sasfies the necessary condions. If your insncts were wrong on some of these, correct your thinking accordingly. The matrix in e) is not in reduced row echelon form since the row of all zeros is not at the boom. The matrix in f) is not in reduced row echelon form since the first 14
1.3 Elementary Row Operaons and Gaussian Eliminaon nonzero entries in rows 2 and 3 are not 1. Finally, the matrix in h) is not in reduced row echelon form since the first entry in column 2 is not zero; the second 1 in column 2 is a leading one, hence all other entries in that column should be 0. We end this example with a preview of what we’ll learn in the future. Consider the matrix in b). If this matrix came from the augmented matrix of a system of linear equaons, then we can readily recognize that the soluon of the system is x 1 = 1 and x 2 = 2. Again, in previous examples, when we found the soluon to a linear system, we were unwingly pung our matrices into reduced row echelon form. . We began this secon discussing how we can manipulate the entries in a matrix with elementary row operaons. This led to two quesons, “Where do we go?” and “How do we get there quickly?” We’ve just answered the first queson: most of the me we are “going to” reduced row echelon form. We now address the second queson. There is no one “right” way of using these operaons to transform a matrix into reduced row echelon form. However, there is a general technique that works very well in that it is very efficient (so we don’t waste me on unnecessary steps). This technique is called Gaussian eliminaon. It is named in honor of the great mathemacian Karl Friedrich Gauss. While this technique isn’t very difficult to use, it is one of those things that is easier understood by watching it being used than explained as a series of steps. With this in mind, we will go through one more example highlighng important steps and then we’ll explain the procedure in detail. . Example 4 .Put the augmented matrix of the following system of linear equaons into reduced row echelon form. −3x 1 − 3x 2 + 9x 3 = 12 2x 1 + 2x 2 − 4x 3 = −2 −2x 2 − 4x 3 = −8 We start by converng the linear system into an augmented ma- S trix. ⎡ ⎣ ⎤ −3 −3 9 12 2 2 −4 −2 ⎦ 0 −2 −4 −8 Our next step is to change the entry in the box to a 1. To do this, let’s mulply row ⎡ 1 1 −3 ⎤ −4 ⎣ 2 2 −4 −2 ⎦ 0 −2 −4 −8 15
Page 1: . . . Fundamentals of Matrix Algebr
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 (a) Give
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Chapter 2 Matrix Arithmec 2.3 Visua
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Chapter 2 Matrix Arithmec Vector Ad
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Chapter 2 Matrix Arithmec Scalar Mu
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Chapter 2 Matrix Arithmec S To draw
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Chapter 2 Matrix Arithmec . Ḋefin
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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 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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