It’s more of a perspective and opinion thing. For example I would not say that an integer is its twos-complement representation, but I am very happy to hold ont

It’s more of a perspective and opinion thing. For example I would not say that an integer is its twos-complement representation, but I am very happy to hold onto that representation as a kind of key for the “real thing”. Similarly, in a vector space there can be abstract vectors, which exist no matter of what basis you choose to write them down. (Choosing a basis = choosing a way to encode them as a 1xN or Nx1 matrix). Often in the theory, general theorems can be proven very efficiently using these abstract vectors, but most of the time when one wants to actually do things to those vectors, picking a convenient basis and working with N coordinates is the best choice.