Suppose $\displaystyle T:V \rightarrow W$ is a linear transformation, where V and W are finite dimensional inner product spaces. Prove that $\displaystyle \text{range } T^{*}=(\text{null } T)^{\perp}$.

I can prove it one direction:

Let $\displaystyle v \in \text{range } T^{*}$.

Then $\displaystyle v=T^{*}w$ for some $\displaystyle w \in W$.

Let $\displaystyle v_1 \in \text{null } T$.

Then $\displaystyle \left<v_1,T^{*}w \right>=\left<Tv_1,w \right>=\left<0,w \right>=0$.

So $\displaystyle \left< v_1, v \right>=0 \, \forall \, v_1 \in \text{null } T$.

Hence $\displaystyle v \in (\text{null } T)^{\perp}$.

We have $\displaystyle \text{range } T^{*} \subseteq (\text{null } T)^{\perp}$.

Can someone show the reverse containment??