A Desmos Lattice for you to play with! Click on it to open in a new page.

You can drag the two basis elements around to see what happens.

I’m not actually drawing all the linear combinations of the basis vectors, only those with fairly small coefficients (orange dots). So you’ll notice that with a “bad” choice of basis you will sometimes see that the lattice elements made from the basis vectors by a small linear combination are not staying particularly near the origin! (For example, try (5,2) and (7.3).)