MATH 601,
AUTUMN 2008
Elementary modifications
1. Definitions.
Let
X
be a finite system of vectors in a vector space
V
over a field
K
. Another such
system
X
is said to arise from
X
by an
elementary modification
if
X
is obtained from
X
by replacing some entry
v
with
(a) a multiple
cv
for some scalar
c
= 0,
or
(b)
v
+
aw
, where
w
occurs in
X
at a different place than the original entry
v
and
a
is a scalar,
and leaving
X
otherwise unchanged. Note that, in this case, if
X
is linearly independent,
then so must be
X
.
Also, it follows that
X
then arises from
X
by an elementary
modification as well.
If, instead of (unordered) systems discussed above, we deal with finite systems that
are
ordered
, we allow a third kind of elementary modifications, in which
X
arises from
X
by switching two neighboring entries, and leaving all remaining entries unchanged.
Given two finite systems
X, X
(ordered or not) in a vector space
V
, we will say that
X
and
X
are
equivalent
, if
X
is obtained from
X
as a result of a finite number of
successive elementary modifications. (The number of these modification may be 0, so that
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 Fall '08
 un
 Linear Algebra, Algebra, Vectors, Vector Space, pj, elementary modiﬁcations

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