finding support of a linear combination of vectors

Question

I need to check how many linear combinations of the vectors inside a vector space of 1 dimensional vectors of size n can be supported on first m-sites or indices 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). How can I do this without brute force manipulations?