(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
2
Homework Solutions
18.335 - Fall 2004
2.1
Count the number of
oating point operations required to compute
the QR decomposition of an m-by-n matrix using (a) Householder
re
ectors (b) Givens rotatio
18.06 Problem Set 2 Solution
Total: 100 points
Section 2.5. Problem 24: Use Gauss-Jordan 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
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?
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
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 th
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
18.06 PSET 8 SOLUTIONS
APRIL 15, 2010
Problem 1. (6.3, #14) The matrix in this question is skew-symmetric (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 de
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
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 = t
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
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
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 corlvolutio
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 5-node graph.
1
2
5
3
4
I have not included edge numbers and arrows. Add them
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
( the
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 @E/@C = 0 and
@E/@D
18.06 Problem Set 4 Solution
Total: 100 points
Section 3.5. Problem 2: (Recommended) Find the largest possible number of independent
vectors among
2
3
1
6 17
v1 = 6 7
405
0
2
3
1
607
v2 = 6 7
4 15
0
2
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
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