I was going through the text Introduction to Algorithms by Cormen et. al. [CLRS] where I came across the following section about matroids.
A matroid is an ordered pair $M = (S, \ell)$ satisfying the following conditions.
$S$ is a finite set.
$\ell$ is a nonempty family of subsets of $S$, called the independent subsets of $S$, such that if $В \in \ell$ and $А \subseteq В$, then $A \in \ell$. We say that $\ell$ is hereditary if it satisfies this property. Note that the empty set $\phi$ is necessarily a member of $\ell$.
If $A \in \ell$, $A \in \ell$, and $|A| < |B|$, then there is some element $x \in В — A$ such that $A\cup\{x\} \in \ell$. We say that $M$ satisfies the exchange property.
Now here in the definition above I was having a doubt regarding the term "independent set/subsets" as I have not encountered this term earlier and I was wondering what could its meaning be. Now then I came to this answer here ,which made me understand that the independence here is quite similar to the concept of linearly independent set of vectors in vector space. But then again,I face a problem here the set $S$ in matroid (assuming each element is a vector) can be over any field $F$ and I have the experience of working over only real field $\mathbb{R}$.
Then in the text there were few example of matroids without proofs such as:
Graphic Matroid: $M_G=(S_G,\ell_G)$ for a graph $G=(V,E)$ where $S_G=E$(set of edges of graph $G$). And they say that a set $A\subseteq S_G$ is independent iff it is acyclic.( I do not quite get how they are getting this "acyclic" condition as equivalent to independence) details of text
In the task scheduling problem the authors say that a set $A$ of tasks is independent if there exists a schedule for these tasks such that no tasks are late.(How are they getting this condition of independence?) details of text
The above two examples are just to indicate the portion where I am facing the difficulty. I would be quite helpful if some could explain me the concept of deriving the independence of sets in the context of matroids for fields other than $\mathbb{R}$ in a hardcore manner with examples. I am facing this problem as I have not been lucky yet to have a formal matroid course.
[CLRS]: Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford, Introduction to algorithms., Cambridge, MA: MIT Press (ISBN 978-0-262-03384-8/hbk; 978-0-262-53305-8/pbk). xix, 1292 p. (2009). ZBL1187.68679.