Linear Algebra, 2020a

Section III. Reduced Echelon Form 61
can derive one from the other. That relation is an equivalence relation, called
row equivalence, and so partitions the set of all matrices into row equivalence
classes.
(
1 3 )
2 7
(
1 3 )
0 1
...
(There are infinitely many matrices in the pictured class, but we’ve only got
room to show two.) We have proved there is one and only one reduced echelon
form matrix in each row equivalence class. So the reduced echelon form is a
canonical form ∗ for row equivalence: the reduced echelon form matrices are
representatives of the classes.
⋆
⋆
⋆
(
1 0 )
0 1
...
⋆
The idea here is that one way to understand a mathematical situation is
by being able to classify the cases that can happen. This is a theme in this
book and we have seen this several times already. We classified solution sets of
linear systems into the no-elements, one-element, and infinitely-many elements
cases. We also classified linear systems with the same number of equations as
unknowns into the nonsingular and singular cases.
Here, where we are investigating row equivalence, we know that the set of all
matrices breaks into the row equivalence classes and we now have a way to put
our finger on each of those classes — we can think of the matrices in a class as
derived by row operations from the unique reduced echelon form matrix in that
class.
Put in more operational terms, uniqueness of reduced echelon form lets us
answer questions about the classes by translating them into questions about
the representatives. For instance, as promised in this section’s opening, we now
can decide whether one matrix can be derived from another by row reduction.
We apply the Gauss-Jordan procedure to both and see if they yield the same
reduced echelon form.
∗ More information on canonical representatives is in the appendix.