Linear Algebra

Section III. Reduced Echelon Form 57
gives this set of equations for i =1uptoi = r.
b1,ℓj
bj,ℓj
br,ℓj
= c1,1d1,ℓj + ···+ c1,jdj,ℓj + ···+ c1,rdr,ℓj
.
= cj,1d1,ℓj + ···+ cj,jdj,ℓj + ···+ cj,rdr,ℓj
.
= cr,1d1,ℓj + ···+ cr,jdj,ℓj + ···+ cr,rdr,ℓj
Since D is in reduced echelon form, all of the d’s in column ℓj are zero except for
dj,ℓj , which is 1. Thus each equation above simplifies to bi,ℓj = ci,jdj,ℓj = ci,j ·1.
But B is also in reduced echelon form and so all of the b’s in column ℓj are zero
except for bj,ℓj , which is 1. Therefore, each ci,j is zero, except that c1,1 =1,
and c2,2 =1,... ,andcr,r =1.
We have shown that the only nonzero coefficient in the linear combination
labelled (∗) iscj,j, which is 1. Therefore βj = δj. Because this holds for all
nonzero rows, B = D. QED
We end with a recap. In Gauss’ method we start with a matrix and then
derive a sequence of other matrices. We defined two matrices to be related if one
can be derived from the other. That relation is an equivalence relation, called
row equivalence, and so partitions the set of all matrices into row equivalence
classes.
All matrices:
✥
✪
✩
✦ ✜
. ( ✢
...
13
27 )
. ( 13
01 )
each class
consists of
row equivalent
matrices
(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.
All matrices:
⋆
⋆
✥
✪
✩
✦ ✜
⋆
✢
⋆
...
( 10
01 )
⋆
one reduced
echelon form matrix
from each class
We can answer questions about the classes by translating them into questions
about the representatives.
∗ More information on canonical representatives is in the appendix.