This preview shows page 1. Sign up to view the full content.
Ribet’s Math 110 Second Midterm, problems and abbreviated solutions
Please put away all books, calculators, and other portable electronic devices—anything with an
ON/OFF switch. You may refer to a single 2sided sheet of notes. When you answer questions,
write your arguments in complete sentences that explain what you are doing: your paper becomes
your only representative after the exam is over.
All vector spaces are ﬁnitedimensional over the ﬁeld of real numbers or the ﬁeld of complex
numbers.
1.
Suppose that
T
is an invertible linear operator on
V
and that
U
is a subspace of
V
that is
invariant under
T
. If
v
is a vector in
V
such that
Tv
∈
U
, show that
v
is an element of
U
.
Quick solution:
Let
u
=
Tv
. Because the restriction of
T
to
U
is invertible, there is a unique
v
±
∈
U
such that
Tv
±
=
u
. Since
Tv
±
=
Tv
and
T
is invertible, we have
v
±
=
v
. Hence we have
v
∈
U
.
2.
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 Davidson,M
 Math

Click to edit the document details