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Every elementary row operation has an elementary inverse, so row equivalence is an equivalence relation
Statement
Every elementary row operation on a finite matrix over a field has an elementary inverse. Consequently row equivalence is reflexive, symmetric and transitive.
Facts & Assumptions
Given: A field and finite matrices of one fixed shape over .
An elementary row operation is a row swap, a scaling of one row by a nonzero field element, or addition of a scalar multiple of one row to another (Elementary row operations and row equivalence for finite matrices over a field).
Proof
A swap is its own inverse; the inverse of is because in a field; and the inverse of is .
The empty operation sequence proves reflexivity, reversing a sequence and replacing every operation by its inverse proves symmetry, and concatenating two finite sequences proves transitivity. These arguments also cover empty matrices and the empty reduction.
Depends on
Used by
- Every elementary matrix is invertible, with inverse given by the reverse elementary operation Corollary
- Row-equivalent matrices have the same row space and the same row rank Lemma
- Elementary row operations on an augmented matrix preserve exactly the solutions of its linear system Theorem
- For equal-sized matrices, row equivalence, equality of row spaces and equality of reduced row echelon forms are equivalent Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 7 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Hefferon, Linear Algebra, 4th ed., Ch. One, §III.1 (standard reference, not scraped)