Hey gummy_ratz.
Will the basis of A span A and thus give you a basis such that all linear combinations of those vector span the set A? If you have a full-rank system, then the basis should provide you with your basis b.
Suppose I have a basis A=(a1,a2,...,an) for the nullspace of a matrix with integer coefficients (i.e. Q-linear combinations of A will give me the span of A). I want to find the integral basis for this (perhaps my terminology isn't correct... but what I mean is I want to find a basis B=(b1,b2,...,bn) where the bi's are integers, such that Z-linear combinations of B will give me the span of the A.)
I'm not sure how to do this. In algebra class, given a field Q(sqrt(d)) I remember finding integral basis' for this... but I can't figure out whether what I'm doing here is at all similar... or much easier... any suggestions would be appreciated!
suppose A = {a_{1},...,a_{n}} is a basis for a vector space V.
suppose further that k ≠ 0 in the field F that V is a vector space over.
is B = {ka_{1},...,ka_{n}} also a basis? let's see:
suppose c_{1}(ka_{1}) +...+ c_{n}(ka_{n}) = 0.
then k(c_{1}a_{1} +...+ c_{n}a_{n}) = 0.
since k ≠ 0, c_{1}a_{1} +...+ c_{n}a_{n} = 0.
by the linear independence of the a_{j}, c_{1} =...= c_{n} = 0.
thus the ka_{j} are also linearly independent.
we know that the a_{j} span V. this means given ANY v in V we can write:
v = c_{1}a_{1} +...+ c_{n}a_{n} for some c_{j} in F.
thus v = (c_{1} /k)(ka_{1}) +....+ (c_{n}/k)(ka_{n}), so the ka_{j} span V as well.
now, suppose that each a_{j} = p_{j}/q_{j}, where the p's and q's are relatively prime integers for each j.
what about k = lcm(q_{1},..,q_{n})? this certainly gives a basis that is all integers.
however, i don't believe you can realize the span of this basis as Z-linear combinations, only as Q-linear combinations.
let me give an example:
suppose we have the matrix:
which has a null space with basis {(8,5,-7)}. now (1,5/8,-7/8) is also in this null space, but NO Z-multiple of (8,5,-7) will ever give us (1,5/8,-7/8).
Hi Deveno,
What I'm looking for is not a basis with integer coefficients that spans the same space as the null space of A... But I want a basis with integer coefficients that hits every vector with integer coefficients in the nullspace of A, i.e. null(A)nZ.
I know that given a basis for my nullspace, I can easily find a basis with integer coefficients by multiplying by a scalar k (e.g. the least common multiple of the the denominators of my generators), but is there a way to make sure that I'm hitting everything in null(A)nZ (the elements in the null space of A which have integer coefficients)?
So in your example, I'm not hoping to hit (1,5/8,-7/8) because it has rational coefficients... but I want to make sure I hit every vector with integer coefficients... (For example, suppose I chose (16, 10, -14) for my basis. Q-linear combinations will give me the span of null(A), but Z-linear combinations will not give me (8,5,-7) ).