Recall from class that a set MRm is a k-dimensional manifold if the following holds: For each x0M there exists a neighborhood x0URm and a (smooth) function F:URmk such that - MU=F1(0), and - the Jacobian matrix DF(x0)R(mk)m has rank mk. Here Rnm denotes the space of all nm matrices. Now define a set M={AR33A has eigenvalues 1=i,2=i, and 3=0 for some >0}, and prove the following. (a) If A0M, then there exists a neighborhood A0UR33 such that if AU then AM if and only if tr(A)=0 and det(A)=0. (b) Defining F:R33=R9R2 by F(A)=(tr(A), det(A)), prove that if A0M then the Jacobian matrix DF(A0)R2R9 has rank 2 . You may use the fact that the eigenvalues of a matrix vary continuously with the matrix, i.e., if matrices A and A are close together, then so are their sets of eigenvalues. Note that (b) implies that M is a 7 -dimensional manifold, that is, has codimension 2 in R33. In part (b) you may regard AR33 as an element of R9, that is, identifya1a4a7a2a5a8a3a6a9R33 with (a1,a2,a3,a4,a5,a6,a7,a8,a9)R9. Hint: Try proving (b) first for the particular choice of matrix 0000000. Recall from class that a set MRm is a k-dimensional manifold if the following holds: For each x0M there exists a neighborhood x0URm and a (smooth) function F:URmk such that - MU=F1(0), and - the Jacobian matrix DF(x0)R(mk)m has rank mk. Here Rnm denotes the space of all nm matrices. Now define a set M={AR33A has eigenvalues 1=i,2=i, and 3=0 for some >0}, and prove the following. (a) If A0M, then there exists a neighborhood A0UR33 such that if AU then AM if and only if tr(A)=0 and det(A)=0. (b) Defining F:R33=R9R2 by F(A)=(tr(A), det(A)), prove that if A0M then the Jacobian matrix DF(A0)R2R9 has rank 2 . You may use the fact that the eigenvalues of a matrix vary continuously with the matrix, i.e., if matrices A and A are close together, then so are their sets of eigenvalues. Note that (b) implies that M is a 7 -dimensional manifold, that is, has codimension 2 in R33. In part (b) you may regard AR33 as an element of R9, that is, identifya1a4a7a2a5a8a3a6a9R33 with (a1,a2,a3,a4,a5,a6,a7,a8,a9)R9. Hint: Try proving (b) first for the particular choice of matrix 0000000