Parallel elements of a matroid ($r(\{x\})=r(\{y\})=r(\{x,y\})=1$)

Distinct nonloops $x,y$ are parallel elements of a matroid when their pair is dependent. Declaring each nonloop parallel to itself makes parallelism an equivalence relation on the nonloops. Transitivity follows directly from augmentation: if $\{a,b\}$ and $\{b,c\}$ are dependent but $\{a,c\}$ independent, the independent singleton $\{b\}$ could not be augmented from the independent pair $\{a,c\}$, contradicting the matroid independence axioms. In a linear matroid, nonzero proportional vectors give parallel elements of a matroid.