From the text:
Section 11.5 (second edition): #1ac, 6, 2ac, 3
  (=page 318-320 first edition, #6 new = #2 old,
#3 new = #5 old. Numbers 1ac, 2ac are written below.)
Section 11.6 #1, (=page 328 #1 first edition). THIS PROBLEM
IS NOW OPTIONAL, AS IS ALL OF SECTION 11.6. SOLUTIONS WILL BE PROVIDED,
AS WELL AS OPTIONAL CLASS NOTES FOR 11.6.
HW1: Recall an affine map f:R^n -->
R^m is given by f(x) = Ax + b, where A is an m by n
matrix, x is an n-vector, and b an m-vector. Prove that for such an
affine map, one has the approximation formula
f(x+h) = f(x) + Ah + E(h),
where in fact the error E(h) is identically zero. Deduce that f
is differentiable, with derivative matrix A.
HW2: Prove that an affine map f(x) = Ax + b from R^n -->
R^n has an inverse function if and only if the matrix A is
non-singular, and that the derivative of the inverse map is A^(-1).
(Hint: if y=Ax+b try to solve for x in terms of y to get the inverse
map.)
For those of you with old editions, here is the text from 1ac, 2ac
above:
1) For each of the following functions, prove that f^(-1) exists and
is differentiable in some nonempty, open set containing (a,b) and compute
D(f^(-1))(a,b).
(a) f(u,v)=(3u-v, 2u+5v), at any (a,b).
(c) f(u,v)=(uv,u^2 + v^2), at (a,b)=(2,5). (Note: this point corresponds
to four possible values of (u,v), i.e. (2,1),(1,2),(-1,-2),(-2,-1),
so there are four possible ``branches'' of the inverse function which
you may choose, and you will get different answers for your corresponding
derivative matrices depending on which branch you pick.
2) For each of the following functions, find out whether the given expression
can be solved for z in a nonempty open set V containing (0,0,0). Is the
solution differentiable near (0,0)?
(a) xyz + sin(x+y+z) = 0.
(c) xyz(2cos(y) - cos(z)) + (zcos(x) - xcos(y))=0.