I am taking a course that mentions that sometime we would like to look at a group $G$ as a group of matrices. From another course I took a while ago I remember that this is called a representation.
I tried looking on Wikipedia, but didn't found a definition, the course I'm taking now is somewhat unformal and said that a representation of a group $G$ is a homomorphism of groups: $\phi:G \to GL_{n}(\mathbb{R})$ or $\phi:G \to GL_{n}(\mathbb{C})$.
Is this the general case ? or maybe we can take any ring ? (any field ?)
This course also claimed (with no proof) :"..then $Im(\phi)\simeq GL_{n}(\mathbb{R})$ " (or $GL_{n}(\mathbb{C}$ in the case of representation to $\mathbb{C}$). Can someone explain this ? (what if $\phi\equiv0 $ ?