I Want to find a basis for $\mathbb Z$-submodule of of $\mathbb Z^3$ satisfing the conditions below:
$x_1+2x_2+3x_3=0 , x_1+4x_2+9x_3=0$
I don't know where to start from. I see that this submodule should be free as $\mathbb Z$ is a P.I.D. and this submodule should be some $M$ isomorphic to $\mathbb Z^3/K$ where K is the Kernel of following homomorphism.
$\mu:\mathbb Z^3 \to M $
how should I form this kernel and find the basis?
