2.3 #9The question is in the pictures Please answer d and e 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 twomatrices H and J from A using a finite sequence of elementary row operations, and bothH and J are in reduced row echelon form, then H = J. Thus, the rref of A is unique. Wewill use the Principle of Mathematical Induction. d.Both H and J below are in rref and have rank 3:1 0 06.71 0 -2 0 7H =0 1 0 -2 4J =0 14 -0 40 0 1 5 -30 0 01-3Explain why the 3rd row of J cannot be expressed as a linear combination of thethree rows of H. Hint: use the fact that the leading 1 is in the 4th column and everyentry to its left is zero.Now, explain in general that the leading 1 in row k of H must be in the samecolumn as the leading 1 in row k of J. Hint: pick the matrix whose leading one inrow k is further to the right.е.

Given:- H=10067010-240015-3 , J=10-207014040001-3

of I MON re now in a position to prove that if A is an m × n mat ces H and J from A using a finite sequence of elementary 1 d J are in reduced row echelon form, then H = J. Thus, the ise the Principle of Mathematical Induction.

of I MON re now in a position to prove that if A is an m × n mat ces H and J from A using a finite sequence of elementary 1 d J are in reduced row echelon form, then H = J. Thus, the ise the Principle of Mathematical Induction.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.2: Direct Methods For Solving Linear Systems

Problem 60EQ

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Question

2.3 #9

The question is in the pictures

Please answer d and e

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.

d.
Both H and J below are in rref and have rank 3:
1 0 0
6.
7
1 0 -2 0 7
H =
0 1 0 -2 4
J =
0 1
4 -0 4
0 0 1 5 -3
0 0 0
1
-3
Explain why the 3rd row of J cannot be expressed as a linear combination of the
three rows of H. Hint: use the fact that the leading 1 is in the 4th column and every
entry to its left is zero.
Now, explain in general that the leading 1 in row k of H must be in the same
column as the leading 1 in row k of J. Hint: pick the matrix whose leading one in
row k is further to the right.
е.

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