Let $f : GL_n (\mathbb{R}) \rightarrow (\mathbb{R^*}, \cdot)$ be a group homomorphism such that $\forall a \in G$
$ f(a) = det(a) $
(a) Describe $Ker(f)$
(b) Describe $Im(f)$
(a) Describe $Ker(f)$
The kernel of $f$ is the set of all matrices whose determinants are $1$ so that $Ker(f) = SL_n(\mathbb{R})$.
(b) The image of $f$ is all of $\mathbb{R^*}$ since for any non-zero real $r \in \mathbb{R^*}$, we have some matrix $a \in GL_n(\mathbb{R})$ s.t. $det(a) = r$.
Is there anything big I'm missing here?