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
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.
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.