= Parallel elements of a matroid
{title2=$r(\{x\})=r(\{y\})=r(\{x,y\})=1$}
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>.
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