In: Math
Show that if P;Q are projections such that R(P) = R(Q) and N(P) = N(Q), then P = Q.
Solution:
Let
where
is a vector space.
Call
the mutual kernel and
the mutual image .
Consider any
and let
and
.
Now,
, since
for all
.
So,
.
where
and
.
, since
and
.
for every
,
Hence,