No injective linear map from $V^*$ to $V$ for infinite dimensional $V$

Question

Suppose $V$ is an infinite dimensional linear space. Show that there is no injective linear map from $V^*$ to $V$.

There is a hint that if $V$ has a basis indexed by $S$, then construct a basis of $V^*$ indexed by $2^S$. But I fail to do so. Can anyone give me some clues?