Vectorspace vs lattice

So given a collection of linearly independent vectors

$$v_1, v_2, \ldots, v_n$$

we can form the n dimensional vector space with the above collection of $v_i$ as our basis. In fact, we basically define the dimension of a Vector space by the size of the smallest set of vectors that span the space. Importantly, given any n linearly independent set of vectors, you can regenerate the entire space. If you have n+1 vectors, you know that you can get an equivalent n vectors from that set of vectors.

This is not the case with Lattices (though you can find a spanning set of dimension n)! To see why, let’s focus on the 1 dimensional case.

The vector (1) will span a lattice of all integers. The lattice spanned by (2) or (3) will not. However, you can recover the original lattice with the span (2) and (3)! If you have (4), and (6) then you can’t even get (1) out of this.