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Kunze Linear Algebra
· Problem 4.3.4
Polynomials
Mathematics — Methods · lagrange-projections-linear-operator
Problem
Let
p
=
(
x
−
2
)
(
x
−
3
)
(
x
−
1
)
p = (x-2)(x-3)(x-1)
p
=
(
x
−
2
)
(
x
−
3
)
(
x
−
1
)
and let
T
T
T
be any linear operator on
R
4
R^4
R
4
such that
p
(
T
)
=
0
p(T) = 0
p
(
T
)
=
0
. Let
P
1
,
P
2
,
P
3
P_1, P_2, P_3
P
1
,
P
2
,
P
3
be the Lagrange polynomials of Exercise 3, and let
E
i
=
P
i
(
T
)
E_i = P_i(T)
E
i
=
P
i
(
T
)
,
i
=
1
,
2
,
3
i = 1, 2, 3
i
=
1
,
2
,
3
. Prove that
E
i
E
j
=
0
if
i
≠
j
,
E
i
2
=
E
i
,
and
T
=
2
E
1
+
3
E
2
+
E
3
.
E_i E_j = 0 \text{ if } i \neq j, \quad E_i^2 = E_i, \quad \text{and} \quad T = 2E_1 + 3E_2 + E_3.
E
i
E
j
=
0
if
i
=
j
,
E
i
2
=
E
i
,
and
T
=
2
E
1
+
3
E
2
+
E
3
.
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