By popular request, I will be posting my lecture notes. Be aware that they are written for myself alone and not for students. Consequently they are badly typeset, they often skip details, etc.
They will be posted here (UW NetID required).
Here's a YouTube series on the "essence of linear algebra". It's remarkably well-done and many people find it helpful for giving the intuition behind the algebra.
While we will cover most of Holt in order, the underlying mathematics is quite old and there are many other expositions. A few:
Here are a couple of row reduction examples.
Here is a summary of the course's material on inverses.
Here is a summary of the course's optional material on diagonalization.