Fundamentals of Matrix Algebra, 2011a

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Chapter 1 Systems of Linear Equaons We have now created a leading 1; that is, the first entry in the first row is a 1. Our next step is to put zeros under this 1. To do this, we’ll use the elementary row operaon given below. −2R 1 + R 2 → R 2 ⎡ 1 1 −3 ⎤ −4 ⎣ 0 0 2 6 ⎦ 0 −2 −4 −8 Once this is accomplished, we shi our focus from the leading one down one row, and to the right one column, to the posion that is boxed. We again want to put a 1 in this posion. We can use any elementary row operaons, but we need to restrict ourselves to using only the second row and any rows below it. Probably the simplest thing we can do is interchange rows 2 and 3, and then scale the new second row so that there is a 1 in the desired posion. R 2 ↔ R 3 ⎡ − 1 R2 → R2 2 ⎡ 1 1 −3 ⎤ −4 ⎣ 0 −2 −4 −8 ⎦ 0 0 2 6 1 1 −3 ⎤ −4 ⎣ 0 1 2 4 ⎦ 0 0 2 6 R3 → R3 We have now created another leading 1, this me in the second row. Our next desire is to put zeros underneath it, but this has already been accomplished by our previous steps. Therefore we again shi our aenon to the right one column and down one row, to the next posion put in the box. We want that to be a 1. A simple scaling will accomplish this. 1 2 ⎣ 0 1 2 4 ⎦ ⎡ 1 1 −3 ⎤ −4 0 0 1 3 This ends what we will refer to as the forward steps. Our next task is to use the elementary row operaons and go back and put zeros above our leading 1s. This is referred to as the backward steps. These steps are given below. ⎡ ⎤ 3R 3 + R 1 → R 1 −2R 3 + R 2 → R 2 −R 2 + R 1 → R 1 1 1 0 5 ⎣ 0 1 0 −2 ⎦ 0 0 1 3 ⎡ 1 0 0 7 ⎤ ⎣ 0 1 0 −2 ⎦ 0 0 1 3 It is now easy to read off the soluon as x 1 = 7, x 2 = −2 and x 3 = 3. . 16
1.3 Elementary Row Operaons and Gaussian Eliminaon We now formally explain the procedure used to find the soluon above. As you read through the procedure, follow along with the example above so that the explanaon makes more sense. Forward Steps 1. Working from le to right, consider the first column that isn’t all zeros that hasn’t already been worked on. Then working from top to boom, consider the first row that hasn’t been worked on. 2. If the entry in the row and column that we are considering is zero, interchange rows with a row below the current row so that that entry is nonzero. If all entries below are zero, we are done with this column; start again at step 1. 3. Mulply the current row by a scalar to make its first entry a 1 (a leading 1). 4. Repeatedly use Elementary Row Operaon 1 to put zeros underneath the leading one. 5. Go back to step 1 and work on the new rows and columns unl either all rows or columns have been worked on. If the above steps have been followed properly, then the following should be true about the current state of the matrix: 1. The first nonzero entry in each row is a 1 (a leading 1). 2. Each leading 1 is in a column to the right of the leading 1s above it. 3. All rows of all zeros come at the boom of the matrix. Note that this means we have just put a matrix into row echelon form. The next steps finish the conversion into reduced row echelon form. These next steps are referred to as the backward steps. These are much easier to state. Backward Steps 1. Starng from the right and working le, use Elementary Row Operaon 1 repeatedly to put zeros above each leading 1. The basic method of Gaussian eliminaon is this: create leading ones and then use elementary row operaons to put zeros above and below these leading ones. We can do this in any order we please, but by following the “Forward Steps” and “Backward Steps,” we make use of the presence of zeros to make the overall computaons easier. This method is very efficient, so it gets its own name (which we’ve already been using). 17
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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 . 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 . Example
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Chapter 2 Matrix Arithmec [ ] 1 −
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Chapter 2 Matrix Arithmec Since mat
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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 3 Operaons on Matrices .
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Chapter 3 Operaons on Matrices Find
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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 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 .
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Chapter 3 Operaons on Matrices 1 2
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Chapter 3 Operaons on Matrices . Ke
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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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