Linear Algebra-Uniqueness of rref

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SUMMARY

The discussion centers on proving the uniqueness of a specific reduced row echelon form (rref) matrix. It is established that a nonsingular matrix in R^3 has a unique rref, which is characterized as a diagonal matrix with 1s on the diagonal or as an identity matrix. Additionally, the uniqueness of the rref implies that the column vectors of the nxn matrix are linearly independent. Roni seeks hints on how to formally prove this uniqueness for a specific matrix.

PREREQUISITES
  • Understanding of reduced row echelon form (rref)
  • Knowledge of nonsingular matrices in R^3
  • Familiarity with linear independence of vectors
  • Basic concepts of matrix theory and linear algebra
NEXT STEPS
  • Study the properties of reduced row echelon form (rref) matrices
  • Learn about the implications of nonsingular matrices in linear algebra
  • Explore proofs of linear independence in vector spaces
  • Investigate the relationship between matrix rank and uniqueness of rref
USEFUL FOR

Students studying linear algebra, particularly those preparing for exams on matrix theory and rref properties, as well as educators looking to clarify concepts of matrix uniqueness and linear independence.

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Homework Statement



"Prove a specific rref matrix is unique"

Homework Equations


The Attempt at a Solution



This is all I have on my review sheet for my next exam. I have the proof of the uniqueness of rref but how do you prove a specific rref matrix is unique ?

say that's any nonsingular matrix in R^3
Would appreciate any hints.

Thanks,
Roni.
 
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If the system is unique, we know it is nonsingular. What does a nonsingular rref matrix look like? It is a diagonal matrix with 1s in the diagonals or identity matrix.

Also, if the nxn matrix is unique, the column vectors are linearly independent.
 

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