Annihilator of an image and rank of the transpose

Let \(V\) be a real vector space and \(X\subseteq V\) a subspace. Write \[ V^*=\{f:V\to\mathbb R:f\text{ is linear}\} \] and \[ X^0=\{f\in V^*:f(x)=0\text{ for every }x\in X\}. \] Let \(T:V\to W\) be a linear transformation of finite-dimensional real vector spaces, and let its transpose \(T^t:W^*\to V^*\) be defined by \(T^t(f)=f\circ T\). Prove that \[ (\operatorname{im}T)^0=\ker(T^t) \] and \[ \dim\operatorname{im}T=\dim\operatorname{im}T^t. \]