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I have a doubt on the definition of Lie groups that I would like to clarify.

Let's have the set of functions [tex]G=\{ f:R^2 \rightarrow R^2 \; | \; < f(x),f(y)>=<x,y> \: \forall x,y \in R^2 \}[/tex], that is the set of all linear functions ℝ^{2}→ℝ^{2}that preserve the inner product. Let's associate the operation of composition to the elements of G and we obtain a group [itex](G,\circ)[/itex].

Now in order to say that G is indeed a Lie group we must prove that G is also asmooth manifold. How can I do this if we don't specify aparametrization(e.g. a matrix representation) for the group G ?

And also in that case, wouldn't the definition of G being a smooth manifold depend on the parametrization?

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# Question on definition of Lie groups

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