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I am a little bit confused about why we define a basis of an $n$-dimensional projective space $P(V)$ to be a set of $n+2$ points in general position, rather than a set of $n+1$ points in general position. For vector spaces a basis is supposed to provide (unique) coordinates. In the world of projective spaces we obtain coordinates by looking at representative vectors of points in general position: If $P(V)$ is an $n$-dimensional projective space, then any choice of representative vectors of $n+1$ points $P_1,\dots,P_{n+1}\in P(V)$ in general position forms a basis of $V$ which provides coordinates for $P(V)$ that are unique up to a common scaling factor. Intuitively, I would stop here and call $P_1,\dots,P_{n+1}$ a basis of $P(V)$.

Flying over my lecture notes the definition with $n+2$ points seems to be motivated by the following lemma.

Lemma. Let $P(V)$ be an $n$-dimensional projective space. For every set of $n+2$ points $P_1,\dots,P_{n+2}\in P(V)$ in general position there exist unique (up to a common scaling factor) representative vectors $v_1,\dots,v_{n+2}\in V$ such that $\sum_iv_i=0$.

However, I don't see what the significance of the $(n+2)$-th point is, or how the addition of the $(n+2)$-th point narrows down the family of possible $(n+1)$-point bases of $P(V)$ to some more desirable subfamily of bases of $P(V)$. After all, for any set of $n+1$ points $P_1,\dots,P_{n+1}\in P(V)$ in general position with representative vectors $v_1,\dots,v_{n+1}\in V$ I can simply create a/the $(n+2)$-th point $P_{n+2}:=\bigl[-\sum_{i=1}^{n+1}v_i\bigr]$.

Can someone explain to me why the $(n+2)$-point definition of 'basis' for projective spaces is more desirable than the more intuitive $(n+1)$-point definition?

Cubi73
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  • Please explain how any choice of $n+1$ points in $P(V)$ provides coordinates for $P(V)$. How does it provide unique (up to a scaling factor) coordinates? – Somos Nov 23 '20 at 01:46
  • @Somos Not the points provide unique (up to scale) coordinates, but any choice of representative vectors of $n+1$ points in general position. Those representative vectors form a basis of $V$ providing unique coordinates for vectors in $V$, which we can use as coordinates of points in $P(V)$. Since points in $P(V)$ are 1-dimensional subspaces of $V$, any two representative vectors of a point are multiples of one another and their uniquely determines coordinates (w.r.t. the representative vector basis of $V$) only differ by a common scaling factor. – Cubi73 Nov 23 '20 at 02:12

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Reading further through my lecture notes I strongly suspect that we prefer bases with $n+2$ points over bases with $n+1$ points, because two sets of $n+2$ points in general position uniquely determine a projective transformation between the corresponding $n$-dimensional projective spaces:

Proposition. Let $P(V)$ and $P(W)$ be $n$-dimensional projective spaces. Any two sets of points $P_1,\dots,P_{n+2}\in P(V)$ and $Q_1,\dots,Q_{n+2}\in P(W)$ in general position uniquely determine a projective transformation $f\colon P(V)\to P(W)$ with $f(P_i)=Q_i$ for all $i=1,\dots,n+2$.

If we just take two sets of $n+1$ points, then any choice of representative vectors uniquely determine a vector space isomorphism $F\colon V\to W$ which uniquely determines a projective transformation $f\colon P(V)\to P(W)$. But this depends on the choice of representative vectors, whereas for $n+2$ points the resulting projective transformation only depends on the $n+2$ points and not their representative vectors.

Cubi73
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