I found that studying linear algebra by getting into vector spaces and linear transformations first made things very easy. This is the approach Halmos or Axler, just to name a few, take.
IMHO, the alternative of going through solving systems of linear equations beforehand, obscures the topic.
What would be equivalent, slightly abstract approaches to study analysis for a freshman? A bit of topology and metric spaces, like Rudin? Multivariate calculus with differential forms, like Hubbard & Hubbard?