I think I might know how to start this problem but I'm not sure how to finish. Here is the statement:
Determine a basis for the ℤ-module of integer solutions to the following system of equations:
$4x + 7y + 2z = 0$
$2x + 4y + 6z = 0$
This is how I was planning on proceeding. First consider the matrix:
$\left( \begin{array}{ccc} 4 & 7 & 2 \\ 2 & 4 & 6 \end{array} \right)$
This represents the system. We can now put this matrix into Smith Normal Form using integer row and column operations. Doing this we get:
$\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \end{array} \right)$
We know this is in SNF because the only nonzero entries are on the diagonal and 1 divides 2 divides 0.
Now I'm having trouble interpreting this matrix though. $z$ should be a free variable, but that's all I'm getting from this. Could anyone give me some hints or tell me what I'm doing wrong?
Dividing the second row by $2$ gives you $$x=-2y-3z$$ while taking $2\cdot \mbox{second row} - \mbox{first row}$ gives you $$y=-10z$$ so that $x=17z$.
So your space solution is generated by $(17,-10,1)$, which is also a $\Bbb{Z}$-basis of the space.