What do you mean with correlation and orthogonality? Like with signal processing, you might calculate the cross-correlation of two signals, and it basically tells you at each possible shifted value, to what extent does one signal project onto the other (so what's their dot product). Orthogonality is not invariant under permuting/shifting entries in just one of the vectors, obviously (e.g. in your standard 2-d arrows space, x-hat is orthogonal to y-hat but not x-hat).
Linear algebra studies linearity, not (just) orthogonality. Orthogonality requires an inner product, and there isn't a canonical one on a linear structure, nor is there any one on e.g. spaces over finite fields. Mathematics, like programming, has an interface segregation principle. By writing implementations to a more minimal interface, we can reuse them for e.g. modules or finite spaces. It also makes it clear that questions like "are these orthogonal" depend on "what's the product", which can be useful to make sense of e.g. Hermite polynomials, where you use a weighted inner product.
Linear algebra studies linearity, not (just) orthogonality. Orthogonality requires an inner product, and there isn't a canonical one on a linear structure, nor is there any one on e.g. spaces over finite fields. Mathematics, like programming, has an interface segregation principle. By writing implementations to a more minimal interface, we can reuse them for e.g. modules or finite spaces. It also makes it clear that questions like "are these orthogonal" depend on "what's the product", which can be useful to make sense of e.g. Hermite polynomials, where you use a weighted inner product.