In Linear Algebra, what is a vector? I understand that a vector space is a collection of vectors that can be added and scalar multiplied and satisfies the 8 axioms, however, I do not know what a vector is. 
I know in physics a vector is a geometric object that has a magnitude and a direction and it computer science a vector is a container that holds elements, expand, or shrink, but in linear algebra the definition of a vector isn't too clear. 
As a result, what is a vector in Linear Algebra?

When I was 14, I was introduced to vectors in a freshman physics course (algebra based). We were told that it was a quantity with magnitude and direction. This is stuff like force, momentum, and electric field.
Three years later in precalculus we thought of them as "points," but with arrows emanating from the origin to that point. Just another thing. This was the concept that stuck until I took linear algebra two more years later.
But now in the abstract sense, vectors don't have to be these "arrows." They can be anything we want: functions, numbers, matrices, operators, whatever. When we build vector spaces (linear spaces in other texts), we just call the objects vectors - who cares what they look like? It's a name to an abstract object.
For example, in $\mathbb{R}^n$ our vectors are ordered n-tuples. In $\mathcal{C}[a,b]$ our vectors are now functions - continuous functions on $[a, b]$ at that. In $L^2(\mathbb{R}$) our vectors are those functions for which
$$ \int_{\mathbb{R}} | f |^2 < \infty $$
where the integral is taken in the Lebesgue sense.
Vectors are whatever we take them to be in the appropriate context.
