I'm looking towards using sets of vectors which span the null space of a matrix to conceptually interpret an aspect of the transformation encoded by the matrix.
In this regard, a basis for the null space can be easily constructed using procedures like reduced row echelon factorization or singular value decomposition.
Several articles propose procedures to find sparse "basis". A related question can be seen to be this one.
Are there spanning sets (consider the matrix to be in $\Bbb Q$) which have coefficients in $\{0,1 \}$ in other words, they are "binary" ?
Can this spanning sets, if they exist and there exists a procedure to find one, be found minimizing a certain criteria ? Specifically, sparsity and/or cardinality.