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? 

Although he is talking about topological spaces, the paragraph in Munkres's Topology may still in some sense answer your question and hence is applicable here:

"They wanted, of course, a definition that was as broad as possible,
  so that it would include as special cases all the various examples that were useful in mathematics - euclidean space, infinite-dimensional euclidean space, and function spaces among them - but they also wanted the definition to be narrow enough that the standard theorems about these familiar spaces would hold for topological spaces in general."

