Ribet’s Math 110 Second Midterm, problems and abbreviated solutions
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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.
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This note was uploaded on 01/27/2010 for the course MATH 7330 taught by Professor Davidson,m during the Spring '08 term at LSU.
 Spring '08
 Davidson,M
 Math

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