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Linear Algebra: A Geometric Approach - 2 Edition - Chapter 1.4 - Problem 16
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Solutions for Linear Algebra: A Geometric Approach | 2nd Edition | ISBN: 9781429215213 | Authors: Ted Shifrin, Malcolm Adams 9781429215213

Solution for problem 16 Chapter 1.4

Prove that the reduced echelon form of a matrix is unique, as follows. Suppose B and C

Linear Algebra: A Geometric Approach | 2nd Edition


Problem 16

Prove that the reduced echelon form of a matrix is unique, as follows. Suppose B and C are reduced echelon forms of a given nonzero \(m \times n\) matrix A.

a. Deduce from the proof of Proposition 4.2 that B andC have the same pivot variables.

b. Explain why the pivots of B and C are in the identical positions. (This is true even without the assumption that the matrices are in reduced echelon form.)

c. By considering the solutions in standard form of Bx = 0 and Cx = 0, deduce that B = C.

Accepted Solution
Step-by-Step Solution:

Step 1 of 4

It is given that, and  are the reduced echelon forms of a given nonzero  matrix .

It is known that every leading entry is  for the reduced echelon form.

Also, all the entries of the column above each leading entry are .

To prove that the reduced echelon form of a matrix is unique.

Chapter 1.4, Problem 16 is Solved

Step 2 of 4


Step 3 of 4

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Prove that the reduced echelon form of a matrix is unique, as follows. Suppose B and C