Finding a Basis for a Submodule of Z^3: A Linear Algebra Homework Problem

[pivoxa15]

Homework Statement

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)}"

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.

 

[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.