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homework 4, due wednesday, 5/4 in class, or in Vladimir’s
box by 4:00pm
1. let
V
=
{
f
±
±
f
(
x,y
) is a polynomial of degree
n
with complex coeﬃcients
}
and let
T
:
V
→
V
be the operator
(
Tf
)(
x,y
) =
f
(
y,x
)
.
a. show that every eigenvalue of
T
is 1 or

1, using the fact that
T
2
=
Id
.
b. let
U
=
Range
(
T
+ 1),
W
=
Range
(
T

1). if
f
∈
V
, show that
Tf
=

f
+
g,
g
∈
U
and
Tf
=
f
+
h,
h
∈
W
if you’re stuck, think about the proof of the uppertriangular theorem.
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Unformatted text preview: c. show that V = U ⊕ W , using part b. d. show that U,W are the same as the space of symmetric and antisymmetric polynomials as in the symmetric functions sheet in the documents section. 2. axler chapter 5, numbers 2,6,12,21 1...
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This note was uploaded on 10/05/2011 for the course MATH 3340 taught by Professor Carlsson during the Spring '11 term at Northwestern.
 Spring '11
 Carlsson
 Linear Algebra, Algebra

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