Chapter 3.6, Problem 10E

Rework exercise $9$ with $G=GL(2,ℝ),$

the general linear group of order $2$

over $ℝ$, and $G\text{'}=ℝ$

under addition.

$M_2(ℝ)$

Exercise $9$:

Let $G$

be the additive group of $2\times 2$ matrices over $ℝ$

and $G\text{'}$

the additive group $ℝ$

of real numbers. Define $\varphi :G\to G\text{'}$ by

$\varphi \left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\right)=a+d$.

(This mapping is called the trace of the matrix.) Prove or disprove that $\varphi$

is a homomorphism. If $\varphi$ is a homomorphism, find ker $\varphi$ and decide whether $\varphi$ is an epimorphism or a monomorphism.