Alright, suppose we are given $V$, a finite dimensional inner product space, and a linear map, $T:V \rightarrow V$, with its corresponding adjoint, $T^\star :V \rightarrow V$. I want to show:
$[im(T)]^\perp = ker(T^\star)$.
However I just, for example, start with the left side and go as this:
$[im(T)]^\perp = \{ v \in V | \langle v,T(x) \rangle=0, \forall x \in V \}$
which I start working on as so:
$= \{ v \in V | \langle v,T(x) \rangle =0, \forall x \in V \}$
$= \{ v \in V | \langle T^\star (v),x \rangle =0, \forall x \in V \}$
But now I get stuck and I really just want to show $T^\star(v)=0$ in my set, but cannot manage to find the logic to get there. Any insight?
Best Answer
Isn't $x$ arbitrary? What can you infer from the properties of the (nondegenerate) inner product?