Linear algebra

Since I'm just a student, I won't make conjectures as to courses outside of the one I am taking. My instructor basically has commited the course to pre-abstract algebra, so it's probably more proof-intensive than the average sophomore-level course. However, proof and abstraction is unavoidable when dealing with vector spaces and subspaces. The hardest part I had with the latter proofs (the incredibly simple "prove that a subspace H of a vector space V is also a vector space," for example) was letting go of Rⁿ. For two years, including high school, everything that I've done has been with the reals and with vectors in R² and R³ in multi-variable calculus. Having to give up my precious Rⁿ for the arbitrary V was very hard, but once I did, the proofs just fell into place. Everywhere in calculus we always had graphs with nice numbers and fairly simple geometric representations, but with arbitrary vector spaces, these tools fail in many places. Since those tools fail, new tools must be built. Although I know I will have much more trouble with next semester's group theory course, I can't help but feel that linear algebra has been a good elementary tool-building course.

Needless to say, if you're at all interested in mathematics, I'd recommend you take a linear algebra course.