Prof. wrote something on the board that I didn't understand.
$ \begin{pmatrix} e^x & 0 \\ 0 & e^{-x} \end{pmatrix}\cong (\mathbb{R}^+,\times)\cong (\mathbb{R},+)$
Can anyone help me with this group (I assume the group operation on the left is matrix multiplication) isomorphism?
If you consider the correspondence $$ M_x:=\left( \begin{array}{cc} e^{x}\,\,\, &0\\[15pt] 0 &e^{-x} \end{array} \right)\to x $$ then it is clear that multiplication of matrices corresponds to sums of their images, $$ \left( \begin{array}{cc} e^{x}\,\,\, &0\\[15pt] 0 &e^{-x} \end{array} \right)\cdot \left( \begin{array}{cc} e^{y}\,\,\, &0\\[15pt] 0 &e^{-y} \end{array} \right)=\left( \begin{array}{cc} e^{x+y}\,\,\, &0\\[15pt] 0 &e^{-(x+y)} \end{array} \right)\to (x+y) $$ Also, the identity matrix corresponds to $0$, the addition neutral, and the inverse of $M_x$ is $M_{-x}$. That defines an isomorphism between those matrices with matrix multiplication and the real line with addition.
Also, the real line with addition is isomorphic to the positive reals with multiplication, via the exponential, $x\to e^x$, since $x+y\to e^{x+y}=e^xe^y$, etc.