Math 260
Exam 1
Jerry L. Kazdan
Feb. 9, 2012
12:00 – 1:20
Directions
This exam has 10 questions (10 points each).
Closed book, no calculators or
computers– but you may use one 3
×
5
card with notes on both sides.
Neatness counts
.
1. Which of the following sets are linear spaces?
a)
The points
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 polynomials
p
(
x
) with
1

1
p
(
x
) cos 2
x dx
= 0.
d)
The set of solutions
y
=
y
(
t
) of
y
+ 4
y
+
y
=
x
2

3. [
Note:
You are
not
being asked
to solve this differential equation. You are only being asked a more primitive question.]
2. Let
S
and
T
be linear spaces and
L
:
S
→
T
be a linear map. Say
V
1
and
V
2
are (distinct!)
solutions of the equations
LX
=
Y
1
while
W
is a solution of
LX
=
Y
2
. Answer the following
in terms of
V
1
,
V
2
, and
W
.
a)
Find some solution of
LX
= 2
Y
1

3
Y
2
.
b)
Find another solution (other than
W
) of
LX
=
Y
2
.
3. Say you have
k
linear algebraic equations in
n
variables; in matrix form we write
AX
=
Y
.
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 Spring '12
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 Linear Algebra, Sets, Jerry L. Kazdan

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