Vector Bundles:  Homework 1  (Note that there are some symbols missing which I could not find in html)

1.  Let A be an nxn matrix.      Show that   sum  1/n (At)ⁿ  = e^At convereges for all t.Show that the matrix valued function  X(t) = e^At solves X'(t) = AX(t).   Prove that e^At + e^As = e^A(t+s). Does e^(A+B) = e^A e^B for arbitrary matrices A and B?

Hint:  To show convergence, define a norm |A| which acts appropriately  under multiplication.
2.  Suppose that  A= O D O^(-1)  where D is a diagonal matrix diag(d₁,d₂...d_n) and O is an arbitrary invertible nxn matrix.   Show that e^A= O(diag(e^d₁,e^d₂ ...e^d_n)O^-1.

3.   If V and W are two vector spaces, show that  V* xO W* = (V xOW)*.

4.  Prove that if T:V-->W is a linear transformation between two vector spaces, then there is a basis e₁, e₂,...e_n of V and a basis f₁, f₂,...f _m of W  such that T(e_i) = f_i for i < k and T(e_i) = 0 for   k<i.  Recall that k is the rank of T.

5.  Show that T   Hom(V,V*)  is self adjoint (T=T*)  if and only if ß = i(T) is symmetric (in V* sO V*).

6.Show that T  in
Hom(V,V*) is skew-symmetric (T=-T*) if and only if ß = i(T) is in V*  V*.

7.  Prove that if  ß in   V*  V*  is non-degenerate (this means that Tin Hom (V,V*) is invertible),  then dim V  is even (say 2n), and there is a basis e₁, e₂,...e_n,f₁, f₂,...f _n of V such that  ß(e_k, e_j) =   ß(f_k,f_j) = 0 and ß(e_k ,f _j) =  ð_kj.
8.  A Euclidean inner product or metric on V is a choice of  G in   V* sOV*  such that G(v,v)>0 for all v= 0.  Define    ||v||²  = G(v,v).  Prove the triangle inequality for ||  ||.

9.  Does the set of Euclidean inner products form a sub-space of V* s 0V*?  Prove that
if G and H are Euclidean inner products, then aG+bH is an inner product for a and b non-negative real numbers, one of which is non-zero.  (This is used to construct a Eucldean metric on a manifold).

10.  Prove that VxO(WxOE)= (VxOW)xOE for any vector spaces V,W and E.