Page after page of mathematical insights and delights! I've never had the opportunity to work through it systematically, but have frequently read excerpts and have never been let down. I would expect nothing less from a figure so great as Lax!

It's worth pointing out in the context of this discussion that the book is, by the author's own design, not an introduction to linear algebra. It is a second course that Lax used to teach his advanced undergraduates and beginning graduate students at the Courant Institute. For example, OP with a high school math background will surely be very puzzled by page two, when a linear space is defined as a field 'acting on' a group. Which is, i think, the 'right' way of thinking about the algebraic structure, in the sense that it greatly simplifies all the intricate moving parts of linear algebra. Anyhow, I second your recommendation!