The approaches of Strang and Klein are complementary.

* Klein uses complex numbers to introduce many concepts of vectors, and uses simple graphics programming with Python to motivate the student. The exercises in that chapter of the Klein are brilliant.

* Klein introduces three different "fields" right away: the reals, the complex numbers, and the bool-valued "GF(2)" (a two-valued Galois Field). Strang sticks with the reals.

* The more traditional approach of Strang to Gauss-Jordan elimination is much smoother than Klein's. For this topic, the Klein suffers for lack of straight-up problem drills.