> I wonder if there are any linear algebra books out there that follow this approach of "build the world, one axiom at a time". Basically any math book—so most

> I wonder if there are any linear algebra books out there that follow this approach of "build the world, one axiom at a time". Basically any math book—so most textbooks don't qualify as math books—will do things this way. SUMS (http://www.springer.com/series/3423 http://www.springer.com/series/3423) and most MAA publications (https://www.maa.org/press/books/book-series https://www.maa.org/press/books/book-series) are good places to look. I didn't see one in a glance at the MAA series, but SUMS has Further Linear Algebra (http://www.springer.com/us/book/9781852334253 http://www.springer.com/us/book/9781852334253). I haven't used it personally, but it looks like it has a chance of being the sort of thing you want.