Math 545 Topology and Geometry of Manifolds Winter 2000
Assignment #1
Due 1/12/2000
I. Required problems.
1. Suppose Vand Ware finite-dimensional vector spaces and F:V→Wis any
linear map. The rank of Fis the dimension of its image, and the nullity of Fis
the dimension of its kernel.
(a) Show that there are bases {E1,... ,E
n}for Vand {E0
1,... ,E0
m}for Wwith
respect to which the matrix of Fhas the block form
Ir0
00
,
where Iris the r×ridentity matrix and ris the rank of F.
(b) Prove the rank-nullity law:
rank F+ nullity F=dimV.
2. Let F:Rn→Rmand G:Rm→Rpbe linear maps. Show that
|F(x)|≤|F||x|,
|F◦G|≤|F||G|,
where
|F|=qPi,j(Fj
i)2
and |G|is defined similarly. (Here Fj
iare the matrix entries of Fwith respect to
the standard bases.)
3. Suppose fi:A→Ris a sequence of real-valued continuous functions defined on
asetA⊂Rn.
(a) Prove the Weierstrass M-test: if there exist positive real numbers Misuch
that supA|fi|≤Miand PiMiconverges, then Pificonverges uniformly on
A.
(b) If fi→funiformly on A, prove that fis continuous.
(c) If A=[a1,b
1]×···×[an×bn] is a closed n-dimensional rectangle and fi→f
uniformly on A, prove that
lim
i→∞ ZA
fidx1···dxn=ZA
fdx
1···dxn.
