Show that the following $\phi: \mathbb{R}\to GL_{2}(\mathbb{R})$ is not a homomorphism.

Question

Show that $\phi: \mathbb{R}\to GL_{2}(\mathbb{R})$ defined by $\phi(a) = \begin{bmatrix} 1 &a \\ 1 & 0 \end{bmatrix}$ is not a homomorphism.

I know I can show this by showing it does not preserve the operations, however I want to show this by showing it is not well-defined. Can I just say that since $\phi(0)\notin GL_{2}(\mathbb{R})$, it is not well-defined?

Answer 1

It's a bad question because you can answer it that way. I think you may want to add that as a comment, but also extend the codomain to all matrices and show it's still not a homomorphism.

I should mention the professors always loved to read my homework because of the detail, and when they handed out the answers sometimes they copied mine verbatim. You may be shooting for less effort, in which case your answer should be fine.

Answer 2

No. Asking if a function is a homomorphism does not make sense when it is not well defined.