= Solution
Choose representative vectors $x_1,\ldots,x_{n+1}$ of the first $n+1$ points; they form a <basis>. Write a representative of the last point as $\sum_i a_ix_i$. General position forces each $a_i\ne0$, since otherwise $n+1$ of the points would lie in a hyperplane. Rescale the <basis> vectors to $a_ix_i$. The tuple is then projectively equivalent to the standard frame
$$
[e_1],\ldots,[e_{n+1}],[e_1+\cdots+e_{n+1}].
$$
Apply the same normalization to the second tuple and compose the two transformations to obtain existence. An automorphism fixing the coordinate points is represented by a <diagonal matrix>. Fixing their sum forces all its diagonal entries to agree, so its projective action is the identity. This proves \b[existence and uniqueness]. At $n=1$ it says \b[the <Möbius group> acts sharply triply transitively on the <projective line>]: any ordered triple of distinct points can be sent to any other in exactly one way.
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