My alma mater took an interesting approach in that they dived right into vector spaces right away. At the time we used Elementary Linear Algebra by Nicholson in the prof's custom order, but it seems they've since written an open textbook based on that approach:

https://ruor.uottawa.ca/items/f66a4ede-e276-486c-9067-9621d5...

Really felt like being thrown in the deep end at the time, especially since it was my first exposure to proofs. Very rewarding when it finally started coming together, though.