Question

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

Short answer

It has been proved thathis an injective homomorphism.

Step-by-step solution

Prove that map is injective

Let us suppose that$x\ne y$ .

Then, $h\left(x\right)\ne h\left(y\right)$. (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,$h\left(x\right)h\left(y\right)$

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

Hence, it is a homomorphism.

Conclusion

Hence, his an injective homomorphism.