Show that $\varphi:\mathbb{R}→Gl_2 (\mathbb{R})$ defined by $\varphi(a)=\begin{pmatrix} 1 & a \\ 0 & 1\end{pmatrix}$ is not an isomorphism $\varphi:\mathbb{R}→Gl_2 (\mathbb{R})$  defined by the matrix  $\varphi(a)=\begin{pmatrix} 1 & a \\ 0 & 1\end{pmatrix}$
An isomorphism is a homomorphism that is also bijective.  $\varphi(a)$ is a homomorphism so in order to show it is not an isomorphism I must show it is not 1 to 1 or onto or that it does not have a 2 sided inverse. I am not sure how to show these things. Any help?

Take $A:=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$; then $A\in GL_2(\Bbb R)$ but $\varphi(a)\neq A$ for every $a\in\Bbb R$, thus $\varphi$ is NOT surjective, and in particular it cannot be an isomorphism.
