# Homework Help: Basis for a submodule?

1. May 11, 2007

### pivoxa15

1. The problem statement, all variables and given/known data
How can I find a basis for a submodule of (the Z-module) Z^3 that is generated by the elements {(2,3,1), (3,4,0), (3,4,6) and (5,1,4)}"

3. The attempt at a solution
Would one way be putting each vector as columns in a matrix and row reduce. Except I got a set from the columns which could not even generate the vectors in the set above.

2. May 11, 2007

### pivoxa15

As happens from time to time, I get an idea for a problem while or just after I finish typing it. And rarer does the idea actually turn out to be correct. This time I may have found the solution.

Make a linear combination of these 4 vectors equal 0 and find the coefficients for one of them in terms of the other 4 by making them into row reduced echelon form although always leaving the entries as integers. Hence one vector is made redundant. The remaining 3 form a basis as it is not linearly independent and will span the submodule.