= Linear matroid
A <linear matroid> has a finite <ground set> indexing a family of <vectors> in a <vector space> over a specified <field>. A <subset> of labels is a <matroid independent set> precisely when its <vectors> are <linearly independent>. Labels distinguish repeated <vectors>; the <ground set> is not required to consist of distinct numerical <vectors>. The <matroid bases> correspond to <bases> for the <linear span> of all the supplied <vectors>.
\b[Exchange outline.] The <linear span> of $|I|$ <linearly independent> <vectors> has <dimension> $|I|$. If $|J|>|I|$, not all the <linearly independent> <vectors> indexed by $J$ can lie in that <linear span>. A <vector> outside it can be added to $I$. Heredity follows because a <linear dependence> in a <subset> would also be a <linear dependence> in the full selection.
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