In class on August 30, we discussed a theorem about existence and uniqueness (this is Theorem 1.2.1 in your book). But what about a proof of the theorem? Well...it requires quite a bit of mathematical background; however, the following notes by Jesse Ratzkin (who actually way back in the day I had for abstract algebra!) give a nice background and proof. Just if you're curious: click here
On October 25 there was a handout on the Laplace transform.
In class it was mentioned that there were three distinct ways of solving a particular non homogeneous system. The third way--using Laplace transforms--requires the most linear algebra, so it was not discussed. However, here are ALL THREE WAYS