(1) If A is an n x n matrix, then prove that (A^T)A (i.e., A transpose multiplied by A) is symmetric. (2) Let S be the set of symmetric n x n matrices. Prove that S is a subspace of M, the set of all n x n matrices. (3) What is the dimension of S? (4) Let the function f : M-->S be defined by f(X)=(X^T)X-I. Compute Df(A). (5) Show that Df(A) is onto when A is an orthogonal matrix. (6) Prove that O, the set n x n orthogonal matrices, is a manifold of dimension (n^2-n)/2. (7) Show that the tangent space to O at I is the space of skew-symmetric matrices. Recall that the skew-symmetric matrices satisfy H^T=-H. (8) Is this the same dimension as in (6)? I need to write an easily-readable solution for a freshman-level theoretical calculus/geometry course. Can anyone please help? Thanks.