>f you know how to handle real or complex coordinate vectors and matrices you’re one only isomorphism away (aka choice of basis) from dealing with an “abstract” vector space

No, you aren't. How would you explain that matrices are both linear transformations and vectors? How would explain what a dual space is? How would you understand the properties of the Fourier transformation, which is a mapping between functions, which are also vectors, and itself also is a vector?