Construct a rank-3 matroid using rank-2 flat Let $E$ be a finite set with size bigger or equal to 3. Let $L$ be a collection of subsets of $E$ such that:


*

*$2 \leq |A| < |E|$ for any $A \in L$

*$|A \cap B| \leq 1$ for any $A, B \in L$
Now show that there exists a unique simple rank-3 matroid $M$ on $E$ such that $L$ is exactly the collection of all rank-2 flat. Also, describe the set of basis of $M$ in terms of $L$.
I have difficulty in finding those independent sets and also do not know how to define a rank function such that all elements in $L$ has rank 2. I can come up with an example like $U_{3, 4}$. In $U_{3, 4}$, $L$ is just all the set with size 2 and it is unique up to isomorphism. At least now I know that there is a matroid that can be built up in this way but fail to prove the more general theorem. 
Any responses will be appreciated.
Update_1: Hinted by Joshua, I consider the characterization of flats of a matroid but used the definition from here. This time I have trouble proving the third axiom and also do not know how to make these flats have rank 2.

Because $M$ is supposed to be simple, every singleton $\{e\}$ for $e \in E$ must be a flat, and these are the flats of rank $1$. This shows uniqueness.
The conditions stated on $L$ are not enough to ensure such $M$ exists. Let $E = \{1,2,3,4\}$ and let $L = \{\{1,2\}, \{1,3\}\}$. The sets $E$ and $L$ satisfy your conditions. But, there is no simple matroid $M$ of rank 3 on $E$ with 2-flats exactly $L$. This is because the only flat covering $\{3\}$ is $\{1,3\}$, and $\{1\}$ does not partition $\{1,2,4\}$. 
