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i. In the same spirit as parts (e) and (g), explain in general why row k – 1 of H must be exactly the same as row k – 1 ofJ. Notice that we are working our way up each matrix. Generalize your arguments j. above: show that if we already know that rows i to k of H and J have already been shown to be equal, then row i – 1 of H and J must also be equal. Since we can continue in this fashion until we reach row 1, this completes the proof that all rows of H must be exactly the same as the corresponding row of J. k. Epilogue: In part (e), we focused on the last non-zero row. Suppose we looked at the first rows instead. Both H and J below are in rref and have rank 3. Show that row 1 of J is a linear combination of the three rows of H. This shows that induction cannot begin at the first row. 1 0 -8 0 -9 14 0 -3 0 0 1 2 0 0 0 0 1 5 H = 6 0 1 6 0 = 0 0 0 0 1

i. In the same spirit as parts (e) and (g), explain in general why row k – 1 of H must be exactly the same as row k – 1 ofJ. Notice that we are working our way up each matrix. Generalize your arguments j. above: show that if we already know that rows i to k of H and J have already been shown to be equal, then row i – 1 of H and J must also be equal. Since we can continue in this fashion until we reach row 1, this completes the proof that all rows of H must be exactly the same as the corresponding row of J. k. Epilogue: In part (e), we focused on the last non-zero row. Suppose we looked at the first rows instead. Both H and J below are in rref and have rank 3. Show that row 1 of J is a linear combination of the three rows of H. This shows that induction cannot begin at the first row. 1 0 -8 0 -9 14 0 -3 0 0 1 2 0 0 0 0 1 5 H = 6 0 1 6 0 = 0 0 0 0 1

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 i, j and k

In the same spirit as parts (e) and (g), explain in general why row k – 1 of H must be exactly the same as row k – 1 of J. i. j. above: show that if we already know that rows i to k of H and J have already been shown to be equal, then row i – 1 of H and J must also be equal. Since we can continue in this fashion until we reach row 1, this completes the proof that all rows of H must be exactly the same as the corresponding row of J. k. Epilogue: In part (e), we focused on the last non-zero row. Suppose we looked at the first rows instead. Both H and J below are in rref and have rank 3. Show that row 1 of J is a linear combination of the three rows of H. This shows that induction Notice that we are working our way up each matrix. Generalize your arguments cannot begin at the first row. 1 0 -8 0 -9 1 4 0 -3 0 H 0 1 2 0 J = 0 0 1 6. 0 0 0 1 0 0 0 0 1
Transcribed Image Text:In the same spirit as parts (e) and (g), explain in general why row k – 1 of H must be exactly the same as row k – 1 of J. i. j. above: show that if we already know that rows i to k of H and J have already been shown to be equal, then row i – 1 of H and J must also be equal. Since we can continue in this fashion until we reach row 1, this completes the proof that all rows of H must be exactly the same as the corresponding row of J. k. Epilogue: In part (e), we focused on the last non-zero row. Suppose we looked at the first rows instead. Both H and J below are in rref and have rank 3. Show that row 1 of J is a linear combination of the three rows of H. This shows that induction Notice that we are working our way up each matrix. Generalize your arguments cannot begin at the first row. 1 0 -8 0 -9 1 4 0 -3 0 H 0 1 2 0 J = 0 0 1 6. 0 0 0 1 0 0 0 0 1
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.
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