It is the Lie group given by $GL(n)\times \mathbb{R}^n$.
We have the product
$$ ( A , v ) \left( A _ { 1 } , v _ { 1 } \right) = \left( A A _ { 1 } , A v _ { 1 } + v \right) $$It represents affine transformations of $\mathbb{R}^n$ into itself.
It is a semidirect product
$$ Aff(\mathbb{R}^n)= \mathbb{R}^{n} \rtimes GL\left(n \right) $$We can see it proceeding like for the Euclidean group.
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Author of the notes: Antonio J. Pan-Collantes
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