I believe one of the first theorems you go about proving in linear algebra is that given a vector space of a given dimension you can find a function that preser

I believe one of the first theorems you go about proving in linear algebra is that given a vector space of a given dimension you can find a function that preserves the linear structure to any other vector space of the same dimension. After proving that they just proceed to work with complex numbers since every other (finite dimensional) vector space is the same in that sense.