Solutions to Problems 6
Hint: Dont pay too much attention to my hints, sometimes they are a little othe-cu and may not be very helpfult. An example being the old hint for Problem
6.2!
Problem 6.1 Let H be a separable Hilbert space. Show that K H is compac
PROBLEM SET 1 FOR 18.102 SPRING 2009
SOLUTIONS
RICHARD MELROSE
Full marks will be given to anyone who makes a good faith attempt to answer
each question. The rst four problems concern the little L p spaces lp . Note that
you have the choice of doing every
SOLUTIONS TO PROBLEM SET 4 FOR 18.102 SPRING 2009
WAS DUE 11AM TUESDAY 10 MAR.
RICHARD MELROSE
Just to compensate for last week, I will make this problem set too short and
easy!
1. Problem 4.1
Let H be a normed space in which the norm satis es the paralle
PROBLEM SET 2 FOR 18.102 SPRING 2009
DUE 11AM TUESDAY 24 FEB.
RICHARD MELROSE
I was originally going to make this problem set longer, since there is a missing
Tuesday. However, I would prefer you to concentrate on getting all four of these
questions reall
18.085 Quiz 1
October 4, 2004
Your name is:
Professor Strang
SOLUTIONS
Grading
OPEN BOOK EXAM
1.
2.
3.
4.
Write solutions onto these pages !
Circles around short answers please !
1) (32 pts.)
This problem is about the symmetric matrix
2 1
0
H = 1
2 1
0 1
Solutions to Problem set 7
Problem 7.1 Question:- Is it possible to show the completeness of the Fourier
basis
p
exp(ikx)/ 2
by computation? Maybe, see what you think. These questions are also intended to
get you to say things clearly.
R
(1) Work out the
TEST 2 FOR 18.102: 9:35 10:55 9 APRIL 2009.
WITH SOLUTIONS
For full marks, complete and precise answers should be given to each question
but you are not required to prove major results.
1. Problem
Let H be a separable (partly because that is mostly what I
PROBLEM SET 3 FOR 18.102 SPRING 2009
MY SOLUTIONS.
RICHARD MELROSE
This problem set is also intended to be a guide to what will be on the in-class
test on March 5. In particular I will ask you to prove some of the properties of the
Lebesgue integral, as b
SOLUTIONS TO PROBLEM SET 5 FOR 18.102, SPRING 2009
WAS DUE l l A M TUESDAY 17 MAR.
RICHARD MELROSE
You should be thinking about using Lebesgue's dominated convergence a t several
points below.
Let f : R -+ C be an element of L1( R ) .Define
f (x) x
E [-L,
Solutions to Problem set 10
Problem P10.1 Let H be a separable, innite dimensional Hilbert space. Show
that the direct sum of two copies of H is a Hilbert space with the norm
(23.18)
H
1
2
H 3 (u1 , u2 ) 7 ! (ku1 k2 + ku2 kH ) 2
H
either by constructing a
Solutions to Problem set 8
Problem 8.1 Show that a continuous function K : [0, 1] ! L2 (0, 2) has the
property that the Fourier series of K(x) 2 L2 (0, 2), for x 2 [0, 1], converges
uniformly in the sense that if Kn (x) is the sum of the Fourier series ov
SOLUTIONS
18.085 Quiz 1
Fall 2005
1) (a) The incidence matrix A is 12 by 9. Its 4th row comes from edge 4:
Row 4 of A = [ 0 0 0 0 1 1 0 0 0 ]
(node 5 to node 6)
(b) The 5th column of A indicates edges 3, 4, 9, 10 in and out of node 5:
Column 5 of A = [ 0
18.085 Quiz 1
October 5, 2007
Your PRINTED name is:
1) (39 pts.)
With h =
1
3
Professor Strang
SOLUTIONS
Grading
1
2
3
there are 4 meshpoints 0, 1 , 2 , 1 and displacements u0 , u1 , u2, u3 .
3 3
a) Write down the matrices A0 , A1 , A2 with three rows tha
Solutions to Problem set 9
P9.1: Periodic functions
Let S be the circle of radius 1 in the complex plane, centered at the origin,
S = cfw_z; |z| = 1.
(1) Show that there is a 1-1 correspondence
(21.40) C 0 (S) = cfw_u : S !
cfw_u : R !
(21.41)
, continuou
18.085 Quiz 1
October 2, 2006
Your PRINTED name is:
1) (36 pts.)
Professor Strang
SOLUTIONS
Grading
1
2
3
(a) Suppose u(x) is linear on each side of x = 0, with slopes u (x) = A on
the left and u (x) = B on the right:
Ax for x 0
u(x) =
Bx for x 0
What is
Solutions
18.085 Quiz 1
1) (30 pts.)
VyLLuo=O
Professor Strarlg
October 6, 2003
A systcrn with 2 springs and masses is fixed-free. Constants arc ell c2.
(a) Write dom.11 the rnatriccs A arid K = ATCA
(b) Prove by two tests (pivots, dctcrrninants, indcpcri
18085 FALL 2002 EXAM 1 SOLUTIONS
DETAILED SOLUTIONS
Problem 1
Let x = (tl, .,t,) and b = (bl, .,b,).
Then
and we would like to solve (but can't)
The short answer is that ATA is always positive definite. The long answer is that A is of rank 1only if all tl