= Solution
{c}
A <projective transformation> of $\mathbb P(\mathbb F^{n+1})$ is induced by an invertible <linear map> $A:\mathbb F^{n+1}\to\mathbb F^{n+1}$:
$$
\boxed{\tau([v])=[Av],\qquad A\in\operatorname{GL}_{n+1}(\mathbb F).}
$$
This is well defined because replacing $v$ by a nonzero scalar multiple does not change $[Av]$. Multiplying $A$ by a nonzero scalar also leaves $\tau$ unchanged. Conversely these are precisely the transformations in the <projective linear group> $\operatorname{PGL}_{n+1}(\mathbb F)$. No field automorphism is included in this definition.
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