Rutger Campbell, Disasters in abstracting combinatorial properties of linear dependence

Let E be a finite set and I be a collection of subsets of E. When is there a set of real vectors indexed by E such that I correspond to its linearly independent subsets? In 1935, Whitney introduced matroids using some necessary conditions for this. However, complete characterizations with various techniques are intractable. This remains the case even if it is already known that there is a set of complex vectors indexed by E whose collection of linearly independent subsets corresponds to I.