Ok so you're assuming that that there is a column of 1s at the beginning. This doesn't have to be the case but if we're assuming that it's the case then it makes it a little easier. What have you learning about the hat matrix? How much linear algebra do you know? Do you know how to take the design matrix and construct the hat matrix?
It takes a while to get used to but it actually simplifies things quite a bit. There are still parts of it that aren't natural to me either but it definitely makes more and more sense the more you deal with it.