(d) If the vector b is the sum of the four columns of A, write down the complete solution to
Ax = b.
Answer:
1
2
3
1
1
+ x4 2
x = + x2
1
0
0
1
0
1
2. (11 points) This problem nds the curve y = C + D 2t which gives the best least squares t
to th
2
Homework Solutions
18.335  Fall 2004
2.1
Count the number of
oating point operations required to compute
the QR decomposition of an mbyn matrix using (a) Householder
re
ectors (b) Givens rotations.
2 3
n
ops.
3
2mn2
(a) See Trefethen p. 7475. Answ
18.06 Problem Set 2 Solution
Total: 100 points
Section 2.5. Problem 24: Use GaussJordan elimination on [U I] to
U 1 :
1 a b
1 0
0 1 c x1 x2 x3 = 0 1
U U 1 = I
0 0 1
0 0
Solution (4 points): Row
1 a b
1 0
0 1 c
0 1
0 0 1
0 0
nd the upper triangular
0
0
4
Homework Solutions
18.335  Fall 2004
4.1
Trefethen 11.1
First note that any x 2 Cm can be writtten as x = xR + x? where xR 2 R(A),
R
x? 2 R(A)? : Now since:
R
x?
R
Ay = 0; 8y 2 Cn =) y
cfw_z
A x? = 0; 8y 2 Cn =) A x? = 0
R
R
2R(A)
we have by denition:
18.06 PSET 3 SOLUTIONS
FEBRUARY 22, 2010
Problem 1. (3.2, #18) The plane x 3y z = 12 is parallel to the plane x 3y z = 0 in Problem
17. One particular point on this plane is (12, 0, 0). All points on the plane have the form (ll in the rst
components)
5
Homework Solutions
18.335  Fall 2004
5.1
Trefethen 20.1
)If A has an LU factorization, then all diagonal elements of U are not zero.
Since A = LU implies that A1:k;1:k = L1:k;1:k U1:k;1:k we get that A1:k;1:k is
invertible.
(We prove by induction that
18.06 Quiz 2
April 7, 2010
Professor Strang
Your PRINTED name is:
1
Your recitation number or instructor is
2
3
1 (33 points)
(a) Find the matrix P that projects every vector b in R3 onto the line in the direction of
a = (2, 1, 3).
Solution The general fo
18.06 PSET 8 SOLUTIONS
APRIL 15, 2010
Problem 1. (6.3, #14) The matrix in this question is skewsymmetric (AT = A):
u = cu2 bu3
1
0
c b
du
a u
= c 0
or u = au3 cu1
2
dt
b a 0
u = bu1 au2
3
(a) The derivative of u(t)2 = u2 + u2 + u3 is 2u1 u + 2u2 u + 2u3
7
Homework Solutions
18.335  Fall 2004
7.1
100
Compute the smallest eigenvalue of the
Hij = 1=(i + j
1).
100
Hilbert matrix
(Hint: The Hilbert matrix is also Cauchy. The
C(i; j) = 1=(xi +y j ) is
(xi +y j ). Any submatrix of a Cauchy
determinant of a Cau
18.085 Prof. Strang
Quiz 2 SOLUTIONS
Your name is:
November 7 2003
Grading
1
2
3
Total
1) 36 pts.)
(a) For
d u
dx
= (x
a) with u(0) = u(1) = 0, the solution is linear on
both sides of x = a (graph = triangle from two straight lines).
wrong with the next s
18.06 Problem Set 1 Solutions
Total: 100 points
Section 1.2. Problem 23: The gure shows that cos() = v1 /kvk and sin() =
v2 /kvk. Similarly cos( ) is
and sin( ) is
. The angle is
.
Substitute into the trigonometry formula cos() cos( ) + sin( ) sin() for c
18.06 Quiz 3 Solutions
sor Strang
May 8, 2010
Profes
Your PRINTED name is:
1.
Your recitation number is
2.
3.
1. (40 points) Suppose u is a unit vector in Rn , so uT u = 1 This problem is about the n by n
symmetric matrix H = I
2u uT
(a) Show directly th
18.085Quiz 3
December 9, 2005
SOLUTIONS
Your PRINTED name is:
1) (30 pts.)
Professor Strang
Grading
1
2
3
(a) Solvc this cyclic convolution cyuation for tllc vcctor d. (I would trans
form corlvolutiorl to multiplication.) Notice that c
=
(5,0,0,0)

(1,1
18.085 Quiz 2
November 2 2007
Professor Strang
Your PRINTED name is:
1) 40 pts.)
Grading
1
2
3
This problem is based on a 5node graph.
1
2
5
3
4
I have not included edge numbers and arrows. Add them if you want to:
not needed.
(a) Find AT A for this grap
3
Homework Solutions
18.335  Fall 2004
3.1
Trefethen 10.1
(a) H = I 2vv where kvk = 1: If v u = 0 ( u is perpendicular to v ),
then Hu = u 2vv u = u: So 1 is an eigenvalue with multiplicity n 1
( there are n 1 linearly independent eigenvectors perpendicu
18 06 Problem Set 6 Solutions
Total: 100 points
Section 4 3 Problem 4: Write down E = kAx bk2 as a sum of four squares
the last one is (C + 4D 20)2 . Find the derivative equations @[email protected] = 0 and
@[email protected] = 0. Divide by 2 to obtain the normal equations A Ax =
18 06 Problem Set 5 Solution
Total: points
Section 4 1 Problem 7 Every system with no solution is like the one in problem
6. There are numbers y , , ym that multiply the m equations so they add up to
0 = 1. This is called Fredholms Alternative:
Exactly on
18.06 Quiz 1
March 1, 2010
Professor Strang
Your PRINTED name is:
1
Your recitation number or instructor is
2
3
4
1. Forward elimination changes x = b to a row reduced Rx = d the complete solution is
4
3
6 7
6 7
x=6 0 7+c
4 5
0
2
3
5
3
6 7
6 7
6 7
6 7
6 1