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?
i’m as much with bourbaki as the next guy. but that’s not really how most engineers learn in practice.
as for treating linear maps between finite dimensional spaces as vectors that’s quite straightforward to do in coordinate terms.
again i refer you to the classics like Golub and van Lean that have been reprinted many many times and educated generations.