I am trying to understand the following two Propositions in James Oxley's book "Matroid Theory"
Prop. 6.1.3
Let $M$ be a simple rank-r matroid and $\mathbb F$ be a field. The following statements are equivalent:
(i) $M$ is $\mathbb F$-representable.
(ii) $PG(r - 1, \mathbb F)$ has a finite subset $T$ such that $M \cong PG( r - 1, \mathbb F)|T.$
(iii) For some $m \geq r,$ there is a finite subset $S$ of $PG(m - 1, \mathbb F)$ such that $M \cong PG( m - 1, \mathbb F)|S.$
So, my understanding is that $(ii)$ and $(iii)$ are saying almost the same thing i.e., $(ii)$ is a special case of $(iii)$ when $m = r.$ Am I correct?
Prop. 6.1.4
Let $k$ be a non-negative integer. When $\mathbb F$ is the finite field $GF(q):$
$(i)$ The number of rank-k flats in $PG(r - 1, q)$ is the Gaussian coefficient ${r \brack k}_q.$
$(ii)$ The number of $k$-element independent sets in $PG(r - 1, q)$ is $$\frac{1}{k!}\frac{q^r - 1}{q - 1}\frac{q^r - q}{q - 1} \dots \frac{q^r - q^{k-1}}{q - 1}.$$
$(iii)$ The number of $k$-element circuits in $PG(r - 1, q)$ is 0 for $k <3$ and is $$\frac{1}{k!(q - 1)}(q^r - 1)(q^r - q)\dots (q^r - q^{k-2} \text{ for } k \geq 3.$$
I want to:
Describe $PG(r - 1, q)/I$ for a $k$- element independent set $I$ in $PG(r - 1, q).$
My guess is:
Using the first Prop.(ii),if we denote all the independent sets of $PG(r - 1, q)$ by $\mathcal{I}$ then $PG(r - 1, q)|(\mathcal{I} - I)$ is isomorphic to a simple matroid of rank r - r(I).
I can also say that by (ii) in the second proposition The number of $k$-element independent sets in $PG(r - 1, q)$ is $$\frac{1}{k!}\frac{q^r - 1}{q - 1}\frac{q^r - q}{q - 1} \dots \frac{q^r - q^{k-1}}{q - 1} - 1.$$
I am not sure if what I am saying above is correct or no, could someone explain this to me please?