You're confusing Gibbs-Heaviside vectors with linear algebra. Linear algebra is the study of linear spaces and linear operators between them. Endowing those spaces with a norm or a dot product is usually as far as linear algebra courses go.
I suppose you could say that any linear space equipped with some product on it falls into linear algebra, in which case it would be, but an obscure corner at best. Its sole application is electromagnetism in a non-relativistic formulation. That's a hugely important application if you're a physicist, but compared to the number of applications of linear algebra it's basically negligible.
The inner product is the only one usually taught in linear algebra, and that's because it's central to talking about representations of linear operators in bases.
