I studied engineering with a European-style curriculum, which starts with groups, rings, vector spaces, linear transforms, eigenvalues, etc - and only then proved that matrices can be used to represent any finite dimensional to finite dimensional linear transformation, and mostly went on from there. It did seem roundabout at the time.
But then, everything was using the same terms, and was relatively simple and straightforward. Fourier transforms? Laplace Transforms? They are all just linear transformations. Functional analysis? It's a lot of inner products, hermitian forms and eigensystems but we were familiar with all the properties, so we only concerned outselves with what's different about it (e.g. the spectral theorem). Coding thoery? It's finite fields, we did those in linear algebra, now it's just applications. Stationary distributions on markov processes? It's application of linear algebra+probability, but works mostly the same whether it's discrete (and representable by a matrix) or continuous (when it isn't).
The syllabus for engineering students was basically the same as for math students, except we only covered proofs in class & homework (but did not have to be able to reproduce them in tests), and that we mostly went through a simple logical progression of ideas rather than a historical one (e.g., cauchy's theorem of analytic complex functions was essentially a two-line application of Green's theorem in the plane, and we spent ~30 minutes discussing it; math students spent ~6 weeks proving it in the way Cauchy historically did, progressing from simpler to complex structures)