Question

Prove that the function$g:R*\to GL(2,R)$defined by$g\left(x\right)=\left(\begin{array}{cc}1& 0\\ 0& x\end{array}\right)$ is an injective homomorphism

Short answer

It has been proved thatgis an injective homomorphism.

Step-by-step solution

Prove the map is injective

Let us suppose that $x\ne y$.

Then, $g\left(x\right)\ne g\left(y\right)$. (Since the matrices have different lower right entries)

Hence, it is an injective map.

Prove that it is a homomorphism

Let $x,y\in R*$.

Consider,$g\left(x\right)g\left(y\right)$

$$\begin{gathered} g\left(x\right)g\left(y\right)=\left(\begin{array}{cc}1& 0\\ 0& x\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& y\end{array}\right) \\ =\left(\begin{array}{cc}1& 0\\ 0& xy\end{array}\right) \\ =g\left(xy\right) \end{gathered}$$

Hence, it is a homomorphism

Conclusion

Hence, gis an injective homomorphism.