In the book 'Geometry of Quantum States' the author states that in an $n$-dimensional affine space, select $n+1$ points $x_i$ so that an arbitrary point $x$ can be written as:
$x=\mu_0x_0+\mu_1x_1+.........\mu_nx_n$ , where $\mu_0+\mu_1+....+\mu_n=1$
My doubt is that in any $n$-dimensional space, we can define any point as a sum of $n$ independent coordinates as the dimension of basis is $n$. So why choose $n+1$ points here as one of these points has to be linearly dependent on the others?
I've been working on the same problem. My thoughts goes as follows.
Define line as affine space $ (\mathbb A, V) $ with $ \dim V = 1 $. It can be described (why?) as $ \{P + \alpha v \mid \alpha \in \mathbb K\} $, where $ v $ is a basis of $ V $ and $ P $ is any point. Now we can prove following proposition: if $ P_1 \neq P_2 $, then $ \{P + \alpha v \mid \alpha \in \mathbb K\} = \{\mu_1 P_1 + \mu_2 P_2 \mid \mu_1 + \mu_2 = 1\} $.
For $ \dim V = 2 $ remember that there exist unique plane passing through 3 points, if they are not on the same line (it is crucial; why?). Prove similar proposition for plane — affine space of dimension $ 2 $.
Now $ \dim V = n $. What conditions we have to impose on $ (O, v_1, \dots, v_n) $ and $ (P_1, \ldots, P_{n + 1}) $ to get the equality as earlier? From proof it should be clear why we take exactly $ n + 1 $ points and what conditions should be.