Hello, I have a geometry question.

I need your help.

Thanks a lot.

Thank you so much.

I really appreciate your help.

Problem 3: Consider the determinant function Det : Mn,n -> R. a) Compute the derivative of Det at I. Verify D(Det), (H) = tr(H), i.e. the derivative of the determinant at I is the trace. b) Let GLn = {A E Mn,n : Det(A) + 0}, this is called the General Linear Group (of R). Argue it is a manifold of dimension n2. c) Check that if A E GLn then D(Det) A is an onto linear map. Hint: Recall Det(AB) = Det(A) Det(B). Perhaps use this with the chain rule and (a). d) For which matrices A E Mn,n is D(Det) A and onto linear function? e) Argue that SLn = {A E GLn : Det(A) = 1} is a manifold of dimension n2 - 1. What is the tangent space at the identity matrix? Note: 3(a) provides perhaps the simplest argument for why the trace is a geo- metric invariant of a matrix. By geometric I mean that the trace is invariant under conjugation, i.e. it is independent of the coordinate system one uses to describe alinear transformation. This also is the heart of why the Laplacian is a geometric invariant of functions