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??

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- Nov 5th 2011, 10:46 AMabhishekkgptheorem on adjoint of linear transformation
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?? - Nov 5th 2011, 12:35 PMFernandoRevillaRe: theorem on adjoint of linear transformation
- Nov 5th 2011, 09:24 PMabhishekkgpRe: theorem on adjoint of linear transformation
- Nov 5th 2011, 10:00 PMDevenoRe: theorem on adjoint of linear transformation
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. - Nov 5th 2011, 10:41 PMabhishekkgpRe: theorem on adjoint of linear transformation