Zero Vectors for Vector Spaces other than $R^n$ I understand what a zero vectors is in $R^n$ but I need some help visualising other zero vectors: 
For example, the vector space of all functions $${ y : \mathbb R \rightarrow \mathbb R \ \ | \ y''+xy'+e^xy=0 } $$
Is the zero vector just $z(x)=0$ ? Explicit examples of less obvious vector spaces would be greatly appreciate
ted. 
Another example could be the set of all functions $$y:\mathbb R\rightarrow\mathbb R \ \ | \ y''= 0$$
In this example wouldn't the zero vector be any functions $z(x)=ax+b$ but does this contradict the fact that the zero vector is unique? Or does that fact mean that the set above is NOT a vector space? 
Kind Regards,
Hugh

A vector space is among other things a group with respect to sum. So the zero vector is exactly the zero of the sum, the unique element that can be added to any other element without changing it.
In both your examples the zero is the constant $z(x) = 0$. In the second example, all the functions $z(x)=ax+b$ are in fact in the space (their second derivative is 0) but they are not the zero of the space, as for instance they cannot be doubled without changing.
