Fall 2013 Math 311-501
Exam 2A: Solutions
c 2013 Art Belmonte
Wed, 09/Oct
(b) Find a basis for the column space of A. What
is the rank of A?
The rst and third columns of U form a
basis for the column space of U. Since A
is row equivalent to U, the corres
Fall 2013 Math 311-501
Exam 1B: Solutions
c 2013 Art Belmonte
Fri, 13/Sep
1. Write the following system as an augmented matrix.
Use rref on your calculator to put the matrix into
reduced row-echelon form. Express all solutions of
the system, if any.
2x1 +
Fall 2013 Math 311-501
Exam 2B: Solutions
c 2013 Art Belmonte
Wed, 09/Oct
The reduced row-echelon form of A is
1 5 0 4 3
0 0 1
1 1
. If x N (A),
U=
0 0 0
0 0
0 0 0
0 0
then x1 + 5x2 4x4 + 3x5 = 0 and
x3 + x4 + x5 = 0. So x has the form
5r + 4s 3t
r
s
Fall 2013 Math 311-501
Exam 1A: Solutions
c 2013 Art Belmonte
Fri, 13/Sep
3. Let A be a nonsingular matrix. Use determinants to
show AT A is nonsingular and that det AT A > 0.
Since A is nonsingular, we have det (A) = 0.
Recall that det AT = det (A).
1.
Fall 2013 Math 311-501
Exam 3A: Solutions
c 2013 Art Belmonte
Mon, 11/Nov
6
1 3 5
1
1 2
and b = 1 .
3. Let A =
1
1 3
1
6
1
1
4
(a) Use the QR command on your calculator to
factor A = QR into the product of a matrix Q
with orthonormal columns and a
Fall 2013 Math 311-501
Exam 3B: Solutions
c 2013 Art Belmonte
Mon, 11/Nov
1
1
1. Let x = and y =
1
1
2 1
12
3. Let A = 1 1 and b = 6 .
2 1
18
(a) Use the QR command on your calculator to
factor A = QR into the product of a matrix Q
with orthonormal c
Fall 2013 Math 311-501
Exam 4C: Solutions
c 2013 Art Belmonte
Sat, 07/Dec
3. Determine the surface area of the portion of a
hyperboloid parameterized by
1 + z2 cos ,
q=
6 z 6,
1 + z2 sin ,
z ,
0 2.
NOTE: See your exam for graphics.
The surface area is
1.
Fall 2013 Math 311-501
Exam 4D: Solutions
c 2013 Art Belmonte
Sat, 07/Dec
3. Compute S f dS, where f = x2 ez y2 z
and S is the part of the cylinder parameterized as
q = [3 cos , 3 sin , z], 0 z 4, 0 2.
The surface integral of the scalar eld f is
NOTE: Se