Giuseppe Piano was the first to establish the axioms of vector space, where the group axioms were linked to the axioms proposed regarding the second law, “number multiplication,” with a basic concept of compatibility.
From your cognitive and scientific point of view, can you give an explanation of this compatibility , which is the secret of the composition of this structure ''vector space'', in a way different from the usual explanation ?
Thanks in advance.
The notion "vector space" simply extends, from a geometrical point of view, the behaviour of vectors in the Euclidean plane. What can we do with vectors in the Euclidean plane? We can add two vectors, having all properties of this addition to can say that a structure of additive group can be identified.
We have also the scalar product of two vectors! But here we have a problem! The scalar product of two vectors in the plane is a real number, not a vector! So such operation is not useful, is not internal! Not compatibility with normal properties of operations on algebraic structures!
But we have also the product of a vector with a real number(called scalar) and this product is a vector too! Good job! Now we have compatibility!
Generalizing, we obtain this new structure, called vector space, which is useful both geometrically and algebraically! This is "the secret", nothing more!
On a set E (of vectors) we have addition and we have multiplication with scalars from fields R or C , obtaining a general structure (VECTOR SPACE) extremely appropriate both for linear algebra and for analysis and its branches (functional analysis, operator theory, ...)