We know that $(GL_{2} (\mathbb{R}), ·)$ and ($\mathbb{R}$ − {0}, ·) are a group. Prove that the function det : $GL_{2} (\mathbb{R})$ → $\mathbb{R}$ − {0} is a homomorphism. What is the kernel of this function?
For the first part I denoted $f=det:Gl_{2}(\mathbb{R})$ → $\mathbb{R}-\{0\}$
And then:
Let $A, B \in GL_{2}$
$F(A \cdot B)=det(A\cdot B)=det(A)\cdot det(B)=f(A)\cdot f(B)$
But my problem is with the kernel. I know that if $f:G \rightarrow H$
$kernel f=\{x \in G: f(x)=e_H\}$
Since my 'H' is $\mathbb{R}-\{0\}$ i don't know what is the identity element there. How can i get the kernel?