# Understanding the connection between $PGL(3, K)$ and $AGL(2, K)$

Sorry if this is a trivial question, but I'm having trouble wrapping my head around the connection between these two groups. It seems intuitively clear to me that the group of invertible affine transformations on $$K^2$$, $$AGL(2, K)$$, should be isomorphic to the subgroup of $$PGL(3, K)$$ consisting of projective linear transformations on $$\mathbb{FP}^2$$ that map the line at infinity to itself. However, the group of affine transformations includes translations by an arbitrary element of $$K^2$$, and it's not clear to me what these correspond to in the projective case.

The construction of $$PGL(3, K)$$ that we've been given is as the set of equivalence classes $$GL(3, K)/\sim$$ where $$A\sim B$$ for $$A, B\in GL(3, K)$$ iff $$A=\lambda B$$ for some $$\lambda\in K^{\times}$$. Which equivalence classes correspond to translations in $$AGL(2, K)$$?

• For $M \in GL_2(K), a \in K^2$ then $x \mapsto a+xM$ is in $Aff_2(K)$ and $(a+xM,1) = (x,1) \pmatrix{M & 0\\ a & 1 }$ – reuns May 10 at 19:25
• @reuns Ah, I'm stupid, thank you – that makes sense – Atticus Stonestrom May 10 at 19:30