Linear Algebra Done Right is, without a doubt, one of my favorites in this category. It tends to be taught as an intermediate or graduate level approach, but things clicked for me with Axler's explanations way better than about anything else I've used.
Sadly the last edition turned their beautiful LaTeX typesetting into something really distracting.
Aside from Axler, I wish someone picked up Halmos text and rendered it in a more modern way. It's a really good one, a bit more advanced, but the text is so cramped it's unpleasant to follow.
