Weighted three-vector matroid

Take labels $E=\{a,b,c\}$ and vectors in the real vector space ${\mathbb{R}}^2$:Here is the complete independence check for all eight subsets. The empty set is independent. Each of $\{a\},\{b\},\{c\}$ is independent because its vector is nonzero. Every pair is independent: the determinants of the two-column matrices $[v_a{v}_b]$, $[v_a{v}_c]$, and $[v_b{v}_c]$ are respectively $1$, $1$, and $-1$. The triple is dependent because $v_c-v_a-v_b=0$. ThereforeDeleting elements from any listed subset stays in this list, verifying heredity. Every exchange comparison has sizes $(0,1)$, $(0,2)$, or $(1,2)$. From the empty set, any element of the larger matroid independent set works: this covers three singleton targets and three pair targets. From any singleton to any pair, every element of the pair outside the singleton works, since every resulting pair is independent. This covers all nine singleton-pair comparisons, including disjoint ones. There are no other comparisons: all fifteen ordered size-increasing pairs satisfy exchange.

The three matroid bases are $\{a,b\}$, $\{a,c\}$, and $\{b,c\}$, all of cardinality two, with weights $13$, $14$, and $11$. The matroid greedy algorithm considers $a$, then $c$, then $b$: it admits $a$, admits $c$ because their determinant is nonzero, and rejects $b$ because the triple is dependent. The resulting matroid basis $\{a,c\}$ has weight $8+6=14$, the largest of the three totals. Every singleton can grow, so none is maximal; the three pairs are both maximal and maximum in cardinality.