What does it mean to find an isomorphism? The question reads:

Let $V=<(1,1,0),(1,0,-1),(3,2,-1)>$ and $W=<1+x^2 , 1-x^2 , x+x^2 >.$
  Find an isomorphism $T:V \to W$.

But I have no idea what it means to find an isomorphism even after googling it. Can anyone help me interpret this problem and teach me how I should approach it?
Thanks,
 A: A bit of conceptual understanding is probably necessary first. So a few words on what an isomorphism is in a rather vague sense:
An isomorphism is a map (which is bijective and has various nice properties but we'll leave those aside for a minute) which basically keeps everything the same. That sounds a little bizarre but let me elaborate.
If I consider $\mathbb{R}^2$ and $\mathbb{C}$, I'm imagining two things which look largely the same, right? If I imagine the point $i$ in $\mathbb{C}$ and also imagine the point {$0,1$} in $\mathbb{R}^2$, they're "in the same place". I'm talking about two completely different things here, when I talk about $\mathbb{R}^2$ and $\mathbb{C}$, but somehow there's a sort of correspondence. 
That's because there's an isomorphism between them. The isomorphism in this case is the one which takes a complex number and turns it into a point in $\mathbb{R}^2$. I take the complex number $a + bi$, for any $a$ and $b$ and turn it into the point {$a,b$}. I've re-labelled stuff, pretty much. Notice that the process above can be reversed. I can take any point in $\mathbb{R}^2$ and turn it into a complex number. This is an isomorphism between the complex plane $\mathbb{C}$ and $\mathbb{R}^2$. I'm really horribly skirting around details here but I'm trying quite hard to illustrate the point.
Now, you need to know a bit more technically what an isomorphism needs to do. You have two vector spaces $V$ and $W$, so you're after an isomorphism of vector spaces. This is not the same as an isomorphism of any other sort of thing. An isomorphism of vector spaces is, by definition, a bijection $T:V \rightarrow W$ which preserves addition and scalar multiplication. 
That means $T(u + v) = T(u) + T(v)$ and $T(cv) = cT(v)$ for all $u$,$v$ in $V$ and $c \in \mathbb{F}$ where $\mathbb{F}$ is the field over which your vector space $V$ lives. I'd guess it's $\mathbb{R}$.
You need to find a map $T$ which does all those things. A good thing to know is that a linear map is determined by it's action on a basis (where the linear map here is your thing $T$).
I suggest you realize that both $V$ and $W$ are vector spaces, and work out a basis for both. Then, from there, I suspect that there is a relatively obvious map $T$ to try out (similar to the example I gave with $\mathbb{R}^2$ and $\mathbb{C}$), which you need to check obeys all the rules above.
As a caveat, the "extra details" I've missed out on above are the fact that $\mathbb{C}$ can be viewed as a 2-dimensional vector space over the field 
$\mathbb{R}$, as can $\mathbb{R}^2$, and then we are considering two genuine vector spaces over the same field. We construct the isomorphism by looking at what it does to the bases of $\mathbb{R}^2$ and $\mathbb{C}$.
