How to show that Orthogonal group is subgroup of general linear group GL(V) where V is vector space?
Since general linear group GL(V) group of all linear transformation which are bijective i.e Invertible.
I know that since isometry is injective and also from rank nullity theorem map from V to V has range of full dimension so it is surjective. But I need more formal proof.
Also since to show it is subgroup we have show x*inverse of y belongs to Orthogonal group O(V) by using bilinear form isometry. Please help me out to show it is surjective using bilinear forms and show it is subgroup.