I've been working Linear Algebra a lot recently; I've been using Schaum's Outlines for Linear Algebra to supplement my understanding. Unfortunately, as nice and organized and Schaum's tends to be, the text fails to provide the reader with a conceptual understanding of even the most rudimentary components of linear algebra, e.g. dot products, cross products, projections, etc.
I've already read through the post located at Book Recommendations for Linear Algebra Proofs
Now, I should mention that the above link does not quite satisfy what I am looking for because the author of Schaum's does provide proofs for the theorems he provides, but the theorems are largely computational; perhaps one of the books mentioned at the aforementioned link provide more coneptually ground proofs. Ultimately, I'm looking for Linear Algebra texts that walk the reader through a step-by-step conceptual understanding of the principles of Linear Algebra, should one exist. I suspect that Linear Algebra Done Right may be a good recommendation for what I'm looking for.
Introduction to Linear Analysis by Kreider, Kuller, Ostberg, ... Makes a clear and unified treatment of Linear algebra and differential equations, both ordinary and partial.