Symmetric Matrices and Manifolds Answer Guide

['AQF]

(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.

 

[Anthony]

Surely if you've been asked to provide the answers, shouldn't you be able to come up with solutions?

Or have I missed the point here?

 

['AQF]

Unfortunately, I was trained as an applied mathematician with few abstract or theoretical courses. This is my first year at this job and for the first time I am completely lost. Can you help?

 

[mathwonk]

the first few of these problems follow immediateoly from the definitions of the concepts. so review those definitions.