How would you call the set generated by the linear combination with integer coefficients of a finite set of vectors in R^n? Apparently, a "lattice" requires that the vectors are linearly independent. I am looking for the "correct" name when the latter condition is not required. As an example, how would you call the set in R given by {a*sqrt{2}+b*\pi: a,b integers}?
Its a Z-module of course. Lattice requires discreteness. You cant construct a sphere packing from a dense set of centers.
Patrick's answer is not completely explicit: If you give your 'finite number of vectors' the name S (a set) then what you have in mind is 'the Z-module generated by S'.
For the R-module generated by S (assuming that the 'vectors' are elements of a linear space over R) is often said to be 'the span of S' or span(S). So, if you have to use the construction often and for various vector sets S you could define the notation Z-span(S) to mean the set you want. Then Z-span({sqrt(2),\pi}) would be your example set.
@Patrick
What do you have in mind with your last two sentences? We spoke about 'a finite number of vectors'.
@Ulrich: the dense situation is the example given in the question : in dimension one Z[sqrt(2)] is dense in R but only generated by two elements namely 1 and sqrt(2)
- Badges
- Science topic