Upvoted for suggesting Linear Algebra Done Right.

To get a sense of the approach, read his earlier article Down with Determinants at http://www.axler.net/DwD.html. That article was the first time I felt I really understood a lot of the material I had already theoretically learned and done well at.

On the real analysis side, I learned from Royden and liked it. But my mind definitely heads the analysis way, and what I find easy someone else might not.