support of linear combination of vectors on a fixed set of 'sites'

Question

Let's say I have a vector space of 1-dimensional vectors of size n. I call the indices of the elements inside these one-dimensional vectors as sites. I need to check how many linear combinations of the vectors inside this vector space can be supported on first m-sites i.e. or in other words, the entries in the linear combination can be non-zero only for sites 1 to m where m < n (at least one entry for sites 1 to m is non-zero while all entries for sites m+1 to n are always zero). Can this done in a non-brute force manner?