Dual Lattice

May 2010
We are studying lattices in my mathematical cryptography course and I'm having a hard time wrapping my head around things. It doesn't help that it has been years since I took linear algebra. Anyway, I do not understand dual lattices. I can't seem to really picture what they look like or what they are, and the more I research them the more I get the same basic definition and not much else. This is the problem I am working on, it contains the definition:

dual lattice.jpg
I don't really get what this matrix looks like. From what I have been reading w are integer vectors, just like v, but the dot product of these vectors isn't another vector so what does this matrix look like? And how does that just get me (V^T)^(-1)? Is it because the entries of V are just integers as well, and since it is a basis for L I am just getting linear combinations of the basis? I have a feeling my questions and comments are pretty naive but I just don't see it and I really want to understand. I always do better when I can play with the math, but I can't play with a definition and this isn't in our text. Thank you for taking the time to explain some of this to me.