What is the Mathematical equivalent of the definition of vectors in physics? Physics has an additional requirement to define physical quantities as vectors i.e. if they are invariant under rotation of the coordinate system for example quantities like force, velocity etc.
How does it correspond to the mathematical formulation? Is it a subset of vector spaces? Is it an alternate characterisation to the axioms of vector space and can it be proved from the former axioms?
From wikipedia:
A vector space over a field F is a set V together with two operations that satisfy the eight axioms.
The first operation, called vector addition or simply addition + : V × V → V, takes any two vectors v and w and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. (Note that the resultant vector is also an element of the set V ).
The second operation, called scalar multiplication · : F × V → V， takes any scalar a and any vector v and gives another vector av. (Similarly, the vector av is an element of the set V ).
Please try to respond in layman's terms as I am just starting with university level maths. If the proof involves advanced concepts, I am satisfied with reference to the topic in which I might encounter this in advanced studies.
 A: From the most basic point of view in mathematics a vector is an element of a vector space. Vector spaces are built from these basic elements so there is no other requirement for an object to be called a vector other than it being a member of a vector space.  Generally in physics , vectors are "things" with magnitude and direction. So if you really want the notion of length and direction, you should consider a normed vector space, a vector space where there is a norm(length) defined on it.  In physic, usually people work on the $n$ dimensional real space $\mathbb{R}^n$ or the complex plane  $\mathbb{C}$ , both of which are a normed vector space. $\mathbb{R}^n$  with the usual Euclidean norm $||x||=||(x_1,x_2,....,x_n)||=(\sum_{x=1}^{n}x_i^2)^{\frac{1}{2}}$ is probably the most common vector space you deal with.
Lastly, in physics taking the inner product or dot product of vectors  is commonplace, so if you really want to define an inner product, you should consider an inner product space, a vector space with an inner product defined on it. So the most general definition of vectors I can think of that closely matches with the idea from physics would be elements of an inner product space.  
Wikipedia has nice explanation on Inner product spaces.  See for more detail https://en.wikipedia.org/wiki/Inner_product_space 
