Linear Algebra Problems
Math 504 – 505
Jerry L. Kazdan
Although problems are categorized by topics, this should not be taken very seriously since
many problems fit equally well in several different topics.
Notation:
We occasionally write
M
(
n,
F
) for the ring of all
n
×
n
matrices over the field
F
, where
F
is either
R
or
C
.
Basics
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
integraltext
1
−
1
p
(
x
)
dx
= 0.
e)
The set of solutions
y
=
y
(
t
) of
y
′′
+ 4
y
′
+
y
= 0.
3. Which of the following sets of vectors are bases for
R
2
?
a).
{
(0
,
1)
,
(1
,
1)
}
b).
{
(1
,
0)
,
(0
,
1)
,
(1
,
1)
}
c).
{
(1
,
0)
,
(
−
1
,
0
}
d).
{
(1
,
1)
,
(1
,
−
1)
}
e).
{
((1
,
1)
,
(2
,
2)
}
f).
{
(1
,
2)
}
4. For which real numbers
x
do the vectors: (
x,
1
,
1
,
1), (1
,x,
1
,
1), (1
,
1
,x,
1), (1
,
1
,
1
,x
)
not
form a basis of
R
4
? For each of the values of
x
that you find, what is the dimension
of the subspace of
R
4
that they span?
5. Let
C
(
R
) be the linear space of all continuous functions from
R
to
R
.
a)
Let
S
c
be the set of differentiable functions
u
(
x
) that satisfy the differential equa-
tion
u
′
= 2
xu
+
c
for all real
x
. For which value(s) of the real constant
c
is this set a linear subspace
of
C
(
R
)?
1