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 1. 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 0 1 6 0 1 2 0 6 0 0 0 1 5 H = ; J = 0 0 0 0 1

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 1. 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 0 1 6 0 1 2 0 6 0 0 0 1 5 H = ; J = 0 0 0 0 1

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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