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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 $AGL(2, K)$?

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  • $\begingroup$ 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 } $ $\endgroup$ – reuns May 10 at 19:25
  • $\begingroup$ @reuns Ah, I'm stupid, thank you – that makes sense $\endgroup$ – Atticus Stonestrom May 10 at 19:30

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