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IMC
1997 IMC
4
space mapping
space mapping
Source: IMC 1997 day 2 problem 4
October 22, 2005
linear algebra
linear algebra unsolved
Problem Statement
(a) Let
f
:
R
n
×
n
→
R
f: \mathbb{R}^{n\times n}\rightarrow\mathbb{R}
f
:
R
n
×
n
→
R
be a linear mapping. Prove that
∃
!
C
∈
R
n
×
n
\exists ! C\in\mathbb{R}^{n\times n}
∃
!
C
∈
R
n
×
n
such that
f
(
A
)
=
T
r
(
A
C
)
,
∀
A
∈
R
n
×
n
f(A)=Tr(AC), \forall A \in \mathbb{R}^{n\times n}
f
(
A
)
=
T
r
(
A
C
)
,
∀
A
∈
R
n
×
n
. (b) Suppose in addtion that
∀
A
,
B
∈
R
n
×
n
:
f
(
A
B
)
=
f
(
B
A
)
\forall A,B \in \mathbb{R}^{n\times n}: f(AB)=f(BA)
∀
A
,
B
∈
R
n
×
n
:
f
(
A
B
)
=
f
(
B
A
)
. Prove that
∃
λ
∈
R
:
f
(
A
)
=
λ
T
r
(
A
)
\exists \lambda \in \mathbb{R}: f(A)=\lambda Tr(A)
∃
λ
∈
R
:
f
(
A
)
=
λ
T
r
(
A
)
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