How would you prove that this is a group isomorphism? Let $$M_{a} = \begin{pmatrix} 1 & a & \frac{a^2}{2} \\ 0  & 1 & a \\ 0 & 0 & 1\end{pmatrix}$$
where $a \in \mathbb{R}$
and let the function $\phi : \mathbb{R}\rightarrow G$ where $\phi(a) = M_a$ and $G$ is the set containing $M_a$. 
I've shown this is a group homomorphism under addition in $\mathbb{R}$ and multiplication in $G$, but how would I show it's an isomorphism, if it is?  
I think it is... mainly from imagining in my head that if I specify any value of $a$ and fix it to be say $x$,
then $M_x$ must bring back the value of $x$ (and only $x$) for the inverse function... but I'm not sure if this is flawed or how to actually mathematically show it.

I suppose you meant $\;G=\text{Im}\,\phi\le  GL(3,\Bbb R)\;$ , and to answer your question: you simply have to prove the easy following fact
$$\ker\phi=\{0\}$$
