Suppose is a linear transformation, where V and W are finite dimensional inner product spaces. Prove that .
I can prove it one direction:
Let .
Then for some .
Let .
Then .
So .
Hence .
We have .
Can someone show the reverse containment??
suppose
so , for all w in W.
then , for all w in W.
so Ty = 0, so y is in null(T).
(note: why can we conclude that Ty = 0? well, we can always pick w = Ty, since Ty
is an element of W, and since <Ty,w> = 0 for all w in W, in particular,
<Ty,Ty> = 0, so Ty = 0).
on the other hand, if y is in null(T),
for any w in W,
but
so we conclude that
so .
now take the perp of both sides.