In "A Short Course on Spectral Theory", page 10, William Arveson asserts that the "$ax+b$ group", ie. the group generated by all dilations and translations of the real line, is isomorpic to the group of all (real) $2\times 2$ matrices of the form $$ \begin{bmatrix} a & b\\ 0 & \frac{1}{a}\end{bmatrix}, \,\,\,\,\,\, a>0 \mbox{ and }, b \mbox{ real}$$
It is very easy to check that the $ax+b$ group is isomorphic to the group of all matrices of the form
$$ \begin{bmatrix} a & b\\ 0 & 1\end{bmatrix}, \,\,\,\,\,\, a>0 \mbox{ and }, b \mbox{ real}$$
So these two matrix groups should be isomorphic. Is this correct? Can someone give me the isomorphism? I've tried for a while and can't seem to get it.