I just want to see if this proof is correct. The only question relating to the exercise I'll be referencing is one that seems to correct a fault in the wording.
Exercise $7$ of Section $1.7$ of Dummit and Foote's Abstract Algebra book says, "Prove that in Example $2$ of this section, the group action is faithful.". Once this question's wording is corrected, this translates to the following:
Suppose $V$ is a non-trivial vector space over a field $F$. Show that the group action $\phi:\mathbb{F}^{\times}\times V\rightarrow V$ defined by $\phi(g,v)=gv$ is faithful.
Here's my attempt.
Suppose $g,h\in\mathbb{F}^{\times}$ such that $\sigma_g=\sigma_h$. That is, for any $v\in V$, $gv=hv$. Adding $-hv$ to both sides gives us $gv-hv=(g-h)v=\overline{0}$ where $\overline{0}$ is the zero vector. Since $V$ is non-trivial and this holds for any $v\in V$, it must be that $g-h=0$. Thus, $g=h$ and the permutation representation associated with $\phi$ is injective. Therefore, $\phi$ is faithful.