Math 260, Spring 2012
Jerry L. Kazdan
Problem Set 1
Due:
In class Thursday, Jan. 19.
Late papers will be accepted until 1:00 PM Friday
.
These problems are intended to be straightforward with not much computation.
1. At noon the minute and hour hands of a clock coincide.
a)
What in the first time,
T
1
, when they are perpendicular?
b)
What is the next time,
T
2
, when they again coincide?
2. Which of the following sets are linear spaces?
a)
{
X
= (
x
1
, x
2
, x
3
) in
R
3
with the property
x
1

2
x
3
= 0
}
b)
The set of solutions
x
of
Ax
= 0, where
A
is an
m
×
n
matrix.
c)
The set of 2
×
2 matrices
A
with det(
A
) = 0.
d)
The set of polynomials
p
(
x
) with
1

1
p
(
x
)
dx
= 0.
e)
The set of solutions
y
=
y
(
t
) of
y
+ 4
y
+
y
= 0.
[
Note:
You are
not
being
asked to solve this differential equation. You are only being asked a more primitive
question.]
3. Consider the system of equations
x
+
y

z
=
a
x

y
+ 2
z
=
b
3
x
+
y
=
c
a)
Find the general solution of the homogeneous equation.
b)
If
a
= 1,
b
= 2, and
c
= 4, then a particular solution of the inhomogeneous equa
tions is
x
= 1
, y
= 1
, z
= 1. Find the most general solution of these inhomogeneous
equations.
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 Spring '12
 STAFF
 Math, Vector Space, 1 W, Linear map, Det

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