Dual operator with weak* dense image

In the lecture course of functional analysis I took we saw following theorem of duality:

Theorem of Duality: Let $X,Y$ be real normed vector spaces and $A:X\to Y$ a bounded linear operator. Then it holds that $A$ is injective if and only if $A^{\star}$ has a weak$^\star$ dense image.

And now come my question: In the following of the lecture notes it's said that by this previous theorem if $A$ is bijective then $A^{\star}$ is injective. Why? How can we conclude this from previous theorem?

$\textbf{Hint}$: Prove that $\mathrm{Ker}(A^*)=\mathrm{Range}(A)^{\perp}$, where for a subset $S$ of $X$, $S^\perp$ is the annihilator of $S$, i.e., $$S^\perp=\{f\in X^*: f(x)=0\text{ for all } x\in S\}.$$