A vector is not a list of numbers. Not even a fixed sized list - a tuple. A vector is just an object which you can add to another object by its same kind and scale it by a real number in a sensible way. The real numbers themself are vectors. But everything you can image.

Take two digital heaps of potatoe mash where you can use a drag to to make it bigger using the mouse in a digital mash heap editor. Adding two heaps gives you a bigger one. You can scale it using the scale tool. A perfectly fine vector - no number lists included.

Polynomials are a canonical example as well, but it does not help when the authors say that anything can be a vector, give them as an example of “other” vectors and then proceed by representing the coefficients again as a list of numbers.

A first clear encounter that vectors are not lists of number is when you have to realize that matrices are vectors too. No longer 1 dimensional tuples but 2 dimensional rectangular ones.

The fundamental problem is that we (I) learn linear algebra on the concrete example of tuples of numbers as vectors. We see the first element of the list as the first dimension. We know how to map that list to a point in a space an so on. Doing that constrains our intution to one example and robs us the chance to use the learned concepts for all other things in live where linear algebra would be useful.

One example is the definition of a dimension. Tuples of 3 numbers are 3 dimensions. But a dimension is much more general and has nothing to do with the lengt of tuples. So what is the dimension of the potato mash vectors? The dimension is the number of basis vectors. Basis vectors are the minimal set, which spans the vector space of all possible potatoe mash heaps.

It is fine to learn things by example. But this requires to unlearn a lot of things and unlearn intuition to generalize.