How did mathematicians decide on the axioms of linear algebra So we vector spaces, linear transformations, inner products etc all have their own axioms that they have to satisfy in order to be considered to be what they are. 
But how did we come to decide to include those axioms and not include others? 
For example, why does  this rule hold in inner product spaces $c\langle u,v\rangle=\langle cu,v\rangle$, when my intuition says that it should be $\langle cu,cv\rangle$?
And how did we decide that it was  scalar multiplication and additive were sufficient criteria for something to be a linear map? 

The motivation to the inner product is perhaps most serious point because
linearity axioms are just abstract properties of the space. Without
additional and scalar construction on it the very space is completely
useless; even for mathematics. Hence we need (mathematics, physics and calculus require) to introduce a way of getting the numeric and invariant (base independent) quantities from vectors; like angles and lengths. This entails that we actually want to have an invariant numeric function of elements of the vector space. The first property of the function is to keep the linearity of the space itself. Hence it is called the linear functional. It is primary, not the inner product. The latter object is just a numeric representation of the functional above (Riesz's theorem). This also lead to the notion of a dual space and its elements. More precisely its coordinate represetations provide the computation tool of both the functional and inner product.
