Connection between linearly independent vectors and projective points in general position

Question

I'm trying to understand the connection between the notions of linear independence and general position. I have no background in geometry, so first I'll start with what I know and then I'll pose specific questions, please bear with me and correct me at any point.

Let $q$ be a prime power, $d$ be a nonnegative integer, and $V$ be a $(d+1)$-dimensional vector space over the finite field $F_q$ with $q$ elements. For $v \in V$ denote $$[v] = \left\{ cv \mid c \in F_q, c\neq 0 \right\}.$$ Then the collection of symbols $[v]$ can be seen as the points of the $d$-dimensional projective space PG(d,q). Furthermore, for a $(k+1)$-dimensional subspace $S$ of $V$, the set $$\left\{ [s] \mid s \in S \right\}$$ is a $k$-flat of $PG(d,q)$.

From what I read in pg. 19 of these notes I assume that this definition of the notion of "general position" is correct:

We say that $m$ points in $PG(d,q)$ are in general position if they are not contained in any $(m-2)$-flat.

So, to my understanding, the following statement is correct:

The points $[v_1], \ldots, [v_m]$ of $PG(d,q)$ are in general position if and only if the vectors $v_1, \ldots, v_m$ are linearly independent.

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My proof. The points $[v_1], \ldots, [v_m]$ are in general position iff they are not contained in any $(m-2)$-flat, which is true iff $v_1, \ldots, v_m$ are not contained in any $(m-1)$-dimensional subspace in $V$. This is the same as linear independence of $v_1, \ldots , v_m$.

My questions are: Is the above statement correct? Are the preceding definitions accurate?

PS. The reason for my confusion is that I've read different definitions for "general position" that I don't understand well, as well as discussions were people explain that general position is not equivalent to linear independence (which I thought my statement above implies). While I'm trying to understand and digest things, it would be very helpful to know if I got the above correctly.

Answer 1

The definition from the notes is not exactly correct. In a projective space of dimension $d$ (projective dimension), a set of points is usually considered to be in general position if no $d+1$ are contained in a hyperplane.

So for example if we consider the Fano plane $\mathrm{PG}(2,2)$, the four points associated with the vectors $$(1,0,0),\ (0,1,0),\ (0,0,1),\ (1,1,1)$$ are in general position, because no $3$ are contained in any line.

The definition Massimo gives is rather for affine independence; I'm guessing he referred to Wikipedia for the definition where the writing is slightly confusing. Note that according to Wikipedia, a set of at most $d+1$ points in general position is affinely independent (ie vectors linearly independent); however we can have sets of more than $d+1$ points that are in general position, in which case these two concepts are different.

Note that a collection of linearly independent vectors will define points that are in general position; but the concepts are not equivalent, as shown be the above example (though note that of those four vectors that I gave, any 3 of them form a linearly independent set).