Last Updated：2022/12/24
Since every linear automorphism of V fixes 0 our interest in the transitivity properties of GL(V) is confined to its action on V* = V - {0}. GL(V) is transitive on V* since any two elements of V* may be chosen as the initial members of two ordered bases; it is not in general 2-transitive because there is no linear automorphism taking an independent pair to a dependent pair.
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Since
every
linear
automorphism
of
V
fixes
0
our
interest
in
the
transitivity
properties
of
GL(V)
is
confined
to
its
action
on
V*
=
V
-
{0}.
GL(V)
is
transitive
on
V*
since
any
two
elements
of
V*
may
be
chosen
as
the
initial
members
of
two
ordered
bases;
it
is
not
in
general
2-transitive
because
there
is
no
linear
automorphism
taking
an
independent
pair
to
a
dependent
pair.
