Math 110
Professor K. A. Ribet
Final Exam
May 18, 2005
This exam was a 180-minute exam. It began at 5:00PM. There were 7 problems,
for which the point counts were 8, 9, 8, 7, 8, 7, and 7. The maximum possible
score was 54.
Please put away all books, calculators, electronic games, cell phones, pagers,
.mp3 players, PDAs, and other electronic devices. You may refer to a single
2-sided sheet of notes. Explain your answers in full English sentences as
is customary and appropriate. Your paper is your ambassador when it is
graded. At the conclusion of the exam, please hand in your paper to your
GSI.
1.
Let
T
be a linear operator on a vector space
V
. Suppose that
v
1
, . . . , v
k
are
vectors in
V
such that
T
(
v
i
) =
λ
i
v
i
for each
i
, where the numbers
λ
1
, . . . , λ
k
are distinct elements of
F
. If
W
is a
T
-invariant subspace of
V
that contains
v
1
+
···
+
v
k
, show that
W
contains
v
i
for each
i
= 1
, . . . , k
.
See my solutions for homework set #11.
2.
Assume that
T
:
V
→
W
is a linear transformation between ﬁnite-dimensional
vector spaces over
F
. Show that
T
is 1-1 if and only if there is a linear transfor-
mation
U
:
W
→
V
such that
UT
is the identity map on
V
.
One direction is obvious; if
UT
= 1
V
and
T
(
v
) = 0, then
v
=
U
(
T
(
v
)) = 0, so
that
T
must be injective. The harder direction is to construct
U
when
T
is given
as 1-1. Choose a basis
v
1