The subjects that are reasonably accessible after Calculus and Linear Algebra are:
- Real Analysis.
- Abstract Algebra.
- Elementary Number Theory.
- Elementary Set Theory.
- Elementary Logic.
- Combinatorics and Discrete Mathematics.
- Differential Equations.
Of these, 1-3 and 6 are "popular", so you want to discard them. 8 is popular among "users" of mathematics (engineers, for example).
While Elementary Set Theory and Elementary Logic will probably help with mathematics in general, they are unlikely to give you any particular insight into either calculus or linear algebra. The best possible insight into calculus will be given by Real Analysis; the best possible insight into Linear Algebra will be given by Abstract Algebra. Elementary Number Theory is the source from which many other areas of mathematics sprung, but it will give you no insight into either linear algebra or into calculus (from Number Theory and Real Analysis you can move on to Complex Analysis and Analytic Number Theory, which will likely be useful; from Number Theory you can also move to abstract algebra and from there to Algebraic Number Theory, which would also shed light on the development of a lot of Algebra).
Combinatorics and discrete math are quite fun and interesting, but again there is very little connection with calculus or with linear algebra. Likewise with Geometry.
Topology is a reach, without some real analysis "in the bag" to fall back on, or a lot more experience with abstraction than provided by one year of Linear Algebra.
It's not that real analysis and linear algebra are "independent of other subjects", it's that the gateways from linear algebra and real analysis to those "other subjects" that have intimate connections with them are precisely the subject that you want to avoid and discard because they are "popular." While "the road less traveled" may sound romantic, you may want to avail yourself of 100+ years of experience of mathematicians who have come up with, if not "royal roads", then at least well-traveled roads that help get you to your destination.