Matrix matroid

Matrix matroid --- матричный матроид.

Let [math]\displaystyle{ M }[/math] be an [math]\displaystyle{ m \times n }[/math] matrix over some field [math]\displaystyle{ K }[/math], [math]\displaystyle{ E(M) }[/math] the set of all the column vectors of [math]\displaystyle{ M }[/math], and [math]\displaystyle{ {\mathcal I}(M) }[/math] the family of all the linearly independent sets of column vectors of [math]\displaystyle{ M }[/math], where we assume the empty set [math]\displaystyle{ \emptyset \in {\mathcal I}(M) }[/math]. Then, [math]\displaystyle{ {\mathcal M}(M) = (E(M),{\mathcal I}(M)) }[/math] is a matroid. A matroid obtained in this way is called a matrix matroid (or a linear matroid) and is called (linearly) representable over the field [math]\displaystyle{ K }[/math]. A matroid representable over the field [math]\displaystyle{ GF(2) }[/math] is said to be binary, and one representable over any field, regular.