Given the matrix:
$$ A=\begin{pmatrix} 3 & 1 \\ -1 & 2 \\ \end{pmatrix}. $$
Let $V =\mathbb Z^2$ and $L = AV$. We want to find basis for $V$ and $L$ and draw the sublattice that exhibit the diagonalization.
I found the diagonal form of the matrix to be:
$$\begin{pmatrix} 1 & 0 \\ 0 & -7 \\ \end{pmatrix}$$
I think we need that because the solution says the basis for $V$ is $\{(1,2) , (0,1)\}$ while the basis for $L$ is $\{(1,2),(0,7)\} $.
But it says we can conclude that from the sublattice. So, can someone briefly point out how I can sketch the sublattice?
EDIT:
Also my attempt at drawing it:
The row and column operations you use in computing the Smith normal correspond to invertible matrices $P,Q \in \operatorname{GL}_2(\mathbb{Z})$ such that $$ D = PAQ $$ where $D$ is the diagonal matrix you found. (Since $-1$ is invertible in $\mathbb{Z}$, we can actually take $D$ to be $$ D = \begin{pmatrix} 1 & 0\\ 0 & 7\end{pmatrix} $$ which is what I will use for the rest of the answer.) In this case, I get $$ D = \left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right) \qquad P = \left(\begin{array}{rr} 0 & 1 \\ 1 & -4 \end{array}\right) \qquad Q =\left(\begin{array}{rr} 1 & 2 \\ 1 & 1 \end{array}\right) \, . $$
We can interpret these as change of basis matrices for $V$. (For more on this, see this post or this post.) The change of basis matrix $P$ contains the information we seek: since $P^{-1} = \begin{pmatrix} 4 & 1\\ 1 & 0\end{pmatrix}$, then the set $$ \{v_1, v_2\} = \left\{\begin{pmatrix} 4\\ 1\end{pmatrix}, \begin{pmatrix} 1\\ 0\end{pmatrix}\right\} $$ is a basis for $V$ such that $L = v_1 \mathbb{Z} \oplus 7v_2 \mathbb{Z}$.
As for drawing the sublattice, take a look at pp. 4-5 of this set of notes by Keith Conrad. He draws a lattice and sublattice with respect to so-called unaligned and aligned bases, which I've copied below.
I think this is the sort of picture you have in mind. All right, below is my attempt at drawing the lattice $V$ and its sublattice $L$. The blue parallelograms are the fundamental parallelograms of $V$ and $L$ using my basis, and the red is the same for the book's. This shows quite clearly that both answers are correct.