- #1
ergospherical
- 1,073
- 1,366
- Homework Statement
- As per title
- Relevant Equations
- N/A
I'd like to clarify a few things; the approach is basically just to show that is isomorphic to a subgroup of which is a smooth manifold (since is an open subset of , so its pre-image under the continuous determinant map is a smooth manifold.)
The hint is to consider replacing each with a matrix block . To show this map is a group isomorphism means to show that , right? So I could write (summation implied over repeated suffices) I tried to write the affect of the map on the elements explicitly, i.e. but this becomes a mess to work out . I think it is clear, by considering e.g. a matrix , that it works, but there is surely a better approach?
The hint is to consider replacing each
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