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f. Both H and J below have their leading 1 in row 3 in the same column: 1 0 0 6. 7 1 0 0 -2 9. H 0 1 0 -2 4 0 1 0 3 -6 0 0 1 -3 0 1 5 -2 Explain why the 3rd row of J cannot be expressed as a linear combination of the three rows of H, and similarly, the 3rd row of H cannot be expressed as a linear combination of the three rows of J. Hint: use the fact that the leading ones in rows 1 and 2 of H and J are above zeroes in row 3.

f. Both H and J below have their leading 1 in row 3 in the same column: 1 0 0 6. 7 1 0 0 -2 9. H 0 1 0 -2 4 0 1 0 3 -6 0 0 1 -3 0 1 5 -2 Explain why the 3rd row of J cannot be expressed as a linear combination of the three rows of H, and similarly, the 3rd row of H cannot be expressed as a linear combination of the three rows of J. Hint: use the fact that the leading ones in rows 1 and 2 of H and J are above zeroes in row 3.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2.3 #9

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Please answer f, g and h

9. Proof of Theorem 1.5.3: The Uniqueness of the Reduced Row Echelon Form: We are now in a position to prove that if A is an m × n matrix, and we obtain two matrices H and J from A using a finite sequence of elementary row operations, and both H and J are in reduced row echelon form, then H = J. Thus, the rref of A is unique. We will use the Principle of Mathematical Induction.
Transcribed Image Text:9. Proof of Theorem 1.5.3: The Uniqueness of the Reduced Row Echelon Form: We are now in a position to prove that if A is an m × n matrix, and we obtain two matrices H and J from A using a finite sequence of elementary row operations, and both H and J are in reduced row echelon form, then H = J. Thus, the rref of A is unique. We will use the Principle of Mathematical Induction.
f. Both H and J below have their leading 1 in row 3 in the same column: 1 0 0 6 7 1 0 0 -2 9 H = 0 1 0 -2 4 J : 0 1 0 3 -6 0 0 1 5 -3 0 0 1 5 -2 Explain why the 3rd row of J cannot be expressed as a linear combination of the three rows of H, and similarly, the 3rd row of H cannot be expressed as a linear combination of the three rows of J. Hint: use the fact that the leading ones in rows 1 and 2 of H and J are above zeroes in row 3. Explain in general that row k of H must be exactly the same as row k of J. Now, let us focus on row k – 1. Both H and J below are in rref, both have rank 3, and their 3rd rows are the same: g. h. 1 0 -8 0 7 150 0 7 H = 0 1 3 0 4 0 0 1 0 2 0 0 0 1 -6 0 0 0 1 -6 Explain why the 2nd row of J cannot be expressed as a linear combination of the three rows of H. Note that this includes possibly using the 3rd row of H.
Transcribed Image Text:f. Both H and J below have their leading 1 in row 3 in the same column: 1 0 0 6 7 1 0 0 -2 9 H = 0 1 0 -2 4 J : 0 1 0 3 -6 0 0 1 5 -3 0 0 1 5 -2 Explain why the 3rd row of J cannot be expressed as a linear combination of the three rows of H, and similarly, the 3rd row of H cannot be expressed as a linear combination of the three rows of J. Hint: use the fact that the leading ones in rows 1 and 2 of H and J are above zeroes in row 3. Explain in general that row k of H must be exactly the same as row k of J. Now, let us focus on row k – 1. Both H and J below are in rref, both have rank 3, and their 3rd rows are the same: g. h. 1 0 -8 0 7 150 0 7 H = 0 1 3 0 4 0 0 1 0 2 0 0 0 1 -6 0 0 0 1 -6 Explain why the 2nd row of J cannot be expressed as a linear combination of the three rows of H. Note that this includes possibly using the 3rd row of H.
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