Is it possible to define a linear transformation for every pair of vector spaces I'm trying to figure out if it's possible to define a linear transformation for every pair of n dimensional vector spaces. My thought process has led me to a possible contradiction to this in using the zero vector. Is it possible to map the zero vector to any other vector?
 A: 
Is it possible to map the zero vector to any other vector?

No, that's not possible. It follows from the (definition of) linearity, that the zero vector must be mapped to the zero vector. Let $T:V \to W$ be a linear transformation between vector spaces $V$ and $W$ over some field $K$, then you have for any scalar $\alpha \in K$ and ${\bf 0_V} \in V$:
$$T\left(\alpha \cdot{\bf 0_V}\right)=\alpha \,T\left({\bf 0_V}\right)
\implies T\left({\bf 0_V}\right)=T\left(0 \cdot{\bf 0_V}\right)=0 \,T\left({\bf 0_V}\right)= {\bf 0_W}$$

I'm trying to figure out if it's possible to define a linear transformation for every pair of n dimensional vectors.

I assume you meant to say: define a linear transformation between two arbitrary $n$-dimensional vector spaces (over the same field - see comment by celtschk below). The answer is then yes and the dimensions needn't even be the same... With the notation above, define $T$ as:
$$T : V \to W : {\bf v} \mapsto {\bf 0_W}$$
The transformation that trivially maps every element ${\bf v} \in V$ to the zero vector ${\bf 0_W} \in W$. You can easily verify that this is indeed a linear transformation, though perhaps a boring one :-).
A: If $V$ and $W$ are vector spaces (over the same field, e.g. both real vector spaces) and $B$ is a basis for $V$, then for every function $f:B\to W$, there is a unique linear transformation $T:V\to W$ such that $T(b)=f(b)$ for all $b\in B$.  So yes, it is always possible, and moreover there are $|W|^{\dim(V)}$ ways to do it.
When it comes to specific vectors, you can do whatever you want to the elements of a linearly independent set (such as a basis), but with a linearly dependent set you are restricted by definition of linearity.  If $T$ is linear and $a_1 v_1+a_2v_2+\cdots a_n v_n=0$, then $a_1 T(v_1)+a_2T(v_2)+\cdots a_n T(v_n)=0$.  
A particular example of the above that follows more directly from the definition of linear is that $T(0)=0$ if $T$ is linear.  For the next simplest case, if $v\neq 0$, then $v$ can be mapped anywhere by a linear transformation.  However, you cannot at the same time map $2 v$ wherever you want, because $T(2v)=2T(v)$. 
In the case where $V$ is the zero vector space, its only element gets sent to the zero vector in $W$.  Just for fun, note that in the context of the first sentence, $B$ is the empty set, and the zero map is the unique linear extension of the unique map from $\varnothing$ to $W$.  
