 
Summary: Orientation.
Let n be a positive integer, let m be a positive integer not exceeding n and let V be an ndimensional
vector space.
Associated subspaces. For each m V we let
Ass() = {v V : v = 0}
and note that Ass() is a linear subspace of V which we call the subspace associated to .
Proposition. Suppose m V {0}. Then dim Ass() m. Moreover, if l = dim Ass() > 0 and
v1, . . . , vl is a basis for Ass() then l m and
= v1 . . . vl
for some ml V .
Proof. Suppose l = dim Ass() > 0 and v1, . . . , vn be a basis for V such that v1, . . . , vl is a basis for
Ass(). Write
=
alt(m,n)
v
()v.
For each i = 1, . . . , l we have
0 = vi =
alt(m,n), irng
v
