Why is $\theta\mapsto p_{\theta}$ one-to-one $\iff$ $(n+1)$ functions $\{F_1,\ldots,F_n,1\}$ are linearly independent? If an n-dimensional model $S=\{p_{\theta}| \theta\in\Theta\}$ can be expressed in terms of the functions $\{C,F_1,\ldots,F_n\}$ on a sample space $X$ and a function $\psi\in \Theta$(parameter space) as
$$p(x;\theta)=\exp\left[C(x)+\sum_{i=1}^{n}\theta^{i}F_{i}(x)-\psi(\theta)\right]$$ then we say that $S$ is an exponential family.
My question is , it is known that $\theta\mapsto p_{\theta}$ is one-to-one if and only if the $(n+1)$ functions $\{F_1,\ldots,F_n,1\}$ are linearly independent. But I am not getting this line. Can someone explain it or give some hint?

Rather than using the ill-suited "one-to-one" qualification, which strictly speaking does not apply in this context, one can state that the parameter $\theta$ is identifiable, i.e., that $\theta_1\ne\theta_2$ implies that the corresponding distributions indexed by  $\theta_1$ and $\theta_2$ differ.
The exact result is that the parameter $\theta$ is identifiable if the statistics $F_1$, ..., $F_n$ are linearly independent and the parameter space $\Theta$ is an open set.
The constraint on $\Theta$ being an open set excludes the artificial addition of a statistic $F_{n+1}$ attached with a parameter $\theta^{n+1}$ taking a single possible value, eg $\theta^{n+1}=1$.
To quote from Geyer:

Theorem 1. $\ $ Every exponential family has a minimal representation, and a representation is minimal if and only if neither
of the following conditions hold.
(i) There exists a hyperplane that contains the natural statistic
vector with probability one.
(ii) There exists a hyperplane that contains the natural parameter
space.

and

Theorem 5. A full exponential family is identifiable if and only if condition (i) of Theorem 1 does not hold. Moreover, if $θ$ and $ψ$ are
distinct parameter values corresponding to the same distribution then
$sθ + (1 − s)ψ$ is contained in the full natural parameter space and
corresponds to the same distribution for all real s.
So, if one uses a minimal representation, then the family is
identifiable.

