Meaning of and Types of Vector Spaces Would it be correct to say that a vector space is any set that is consistent with the list of 8 axioms? The 8 axioms being associativity, commutativity, identity element, etc...
In physics we are taught about vectors in the form of arrows. These arrows are sometimes represented by a direction and a magnitude. They are also sometimes represented by their component parts in the form ai + bj + ck. They might also be represented as row or column vectors.
Are all of these things different vector spaces? Or are arrows, magnitudes with directions, vector components, column vectors, and row vectors all different ways to represent the same vector space?
 A: The short answer is in your last paragraph: they are 

different ways to represent the same vector space.

Longer answer:
You seem to be asking about the vectors that come up in beginning physics. You list several ways to think about those vectors, each of which is useful in some context. But there is just one kind of three dimensional real vector. 
The standard mathematical description uses three real coordinates, so a vector is a triple $(x,y,z)$, or, equivalently, $xi + yj + zk$, where $i,j,k$ is a chosen coordinate system. Sometimes you want to think of those triples as rows or columns. Sometimes you want to describe them using their length and two angles.
Vectors as arrows are a little trickier, since the position of the starting point of the arrow matters. For example, you can only add such vectors if they start at the same place.
When you study more advanced mathematics and physics you encounter much more generality. Vector spaces need not have dimension $3$. Often you will want to consider a set of functions as a vector space. Then thinking about arrows or about direction and magnitude don't help as much, and may not even make sense.
