I am trying to figure out what will happen to the uniform matroid $U_{2,m}$ if we remove an element e from it, where e is neither a coloop nor a loop. I am guessing that it will become disconnected but I am not sure from this.
A uniform matroid is defined as: If m and n are non-negative integers with $m \leq n.$ Let $E$ be an $n$-element set and $\mathcal{B}$ be the collection of $m$-element subsets of $E.$ Then it is easy to check that $\mathcal{B}$ is the set of bases of a matroid on $E.$ We denote this matroid by $U_{m,n}$ and call it the uniform matroid of rank $m$ on an $n$-element set.
I also know that the drawing of a $U_{2,5}$ matroid is just a line segment with 5 dots on it representing the vertices from a picture online, is this always the drawing of a uniform matroid $U_{2,m}$ matroid?
Any clarification will be greatly appreciated!
First we should note that $U_{2, n}$ has no loops, since the rank of any singleton is 1. If $n\geq 3$, $U_{2, n}$ has no coloops, since every element is in a circuit (any 3 element set containing said element). Thus, the condition of $e$ not being a loop or coloop is irrelevant unless $n=2$, in which case both elements are coloops. So let $n\geq 3$.
Now, recall that given a matroid $M$ with elements $E$ and an element $e\in E$, the matroid $M\backslash e$ has for independent sets those of $M$ contained in $E-e$, or equivalently, the bases of $M\backslash e$ are the bases of $M$ contained in $E-e$. Hence, $U_{2, n}$ minus an element is isomorphic to $U_{2, n-1}$. Hence, the resulting matroid is not connected if and only if $n=3$. In general, $U_{m,n}$ is connected unless $m\in \{0,n\}$.
Now, if you were to delete more than one element, you are bound to decrease the rank of the uniform matroid at some point, as you at some point will not have enough elements to keep the rank up. In general, for the uniform matroid $U_{m, n}$ and a set $T$ of $t$ elements,
$$ U_{m, n}\backslash T \cong \begin{cases}U_{n-t, n-t}, & \text{if } n-m\leq t\leq n,\\ U_{m, n-t}, & \text{if } t<n-m. \end{cases} $$ For instance, if we delete an element from $U_{2,2}$ (which is necessarily a coloop) we obtain $U_{1,1}$.
Indeed, the drawing of $U_{2, m}$ is in general a line with $m$ points. Lines in such matroid drawings are rank 2 flats.