Gil Strang taught 18.06, the more calculation-focused and elementary version of linear algebra, when I was at MIT. The approach mentioned by the OP was what was taken in the more theoretical version, 18.701. I think that class has notes and lectures on OCW.

Not an online course, but Linear Algebra by Serge Lang is nice little book that is in this spirit (good to look at if you’re familiar with LA basics).

3blue1brown has great linear algebra videos that take it from the transformations point of view.