PROBLEM SET IV
DUE FRIDAY,  MARCH
Exercise . We never proved that derivatives are unique. Show that they are. at is, suppose U ⊂ Rm an open
subset, g : U . Rn a map, and x ∈ U a point. en if φ, ψ : Rm . Rn are two linear maps such that both
|g(x + v) − g(x) − φ(v)|
= 0 and
v→0
|v|
lim
|g(x + v) − g(x) − ψ(v)|
= 0,
v→0
|v|
lim
then φ = ψ.
Recall that the vector space Lin(Rm , Rn ) of linear maps Rm . Rn is isomorphic to the vector space Matn×m (R) ∼
=
mn
m
n

R of n × m matrices. So we may speak of continuity and differentiability for maps to and from Lin(R , R ).
Exercise . Show that the map
K : Lin(Rm , Rn ) × Lin(Rk , Rm ) .
Lin(Rk , Rn )
given by composition (equivalently, the map
Rmn+km ∼
= Matn×m (R) × Matm×k (R) .
Matn×k (R) ∼
= Rkn
given by matrix multiplication) is differentiable at any point (φ, ψ) ∈ Lin(Rm , Rn ) × Lin(Rk , Rm ); compute its
derivative D(φ,ψ) K.
Exercise . Consider the subset GL(Rn ) ⊂ Lin(Rn , Rn ) of the isomorphisms Rn . Rn . (Equivalently, you may
consider the subset GLn (R) ⊂ Matn×n (R) of invertible n×n matrices.) Show that GL(Rn ) is an open set, and prove
that the function I : GL(Rn ) . GL(Rn ) given by I(φ) := φ−1 is differentiable at any point φ ∈ Lin(Rn , Rn );
compute its derivative Dφ I.
Exercise . (A) If M ∈ Matn×n (R), show that the limit
R
∑
1 r
M
lim
R→∞
m!
r=0
exists; call the resulting matrix exp(M).
(B) Prove that if M, N ∈ Matn×n (R) such that MN = NM, then exp(M + N) = exp(M) exp(N) and that
exp(0) = id.
(C) Show that if N is an invertible matrix, then
exp(NMN−1 ) = N exp(M)N−1 .
(D) Show that for any matrix M ∈ Matn×n (R), the matrix exp(M) is invertible.
(E) Finally, show that exp defines a differentiable map Matn×n (R) . GLn (R), and compute its derivative.
Definition. Suppose X ⊂ Rm and Y ⊂ Rm two subsets, and suppose α a nonnegative real number; then a map
g : X . Y is said to be Lipschitz if there is a nonnegative real number M such that for any x, y ∈ X,
|g(x) − g(y)| ≤ M|x − y|.
Recall that we may define an isomorphism M : Lin(Rm , Rn ) .
Matn×m (R) by M(φ) := (φ(e1 ), φ(e2 ), . . . , φ(em )), where we think
of the vectors φ(ei ) arranged vertically.

Exercise⋆ . Suppose U ⊂ Rm a convex open subset, and suppose that g : U .
is differentiable at every point of U. Show that g is Lipschitz if and only if
sup
Rn is a continuous function that
|Dx g(ξ)| < ∞.
x∈U, ξ∈Sm−1
Deduce that if g is continuously differentiable, then its restriction to any compact subset K ⊂ U is Lipschitz.
