Finding a base for a submodule

Finding a base for a submodule

I. Find a base for the submodule of $\mathbb Z^{(3)}$ generated by
$f_1=(1,0,-1)$, $f_2=(2,-3,1)$, $f_3=(0,3,1)$, $f_4=(3,1,5)$.
My question here: Is it enough to find the Smith normal form of the matrix
$A$ consisting of the elements of $f_i$'s or one must find the form
$A'=QAP^{-1}$? If it is the last case, how I can find this form?
II. Find a base for the $\mathbb Z$-submodule of $\mathbb Z^{(3)}$
consisting of all $(x_1,x_2,x_3)$ satisfying the conditions
$x_1+2x_2+3x_3=0$, $x_1+4x_2+9x_3=0$.
How to handle this question?