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Solution
ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-4/19i/c/solution
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Past exam of the mathematics course of the University of Cambridge
/
2019
/
ii
/
Paper 4
/
19I
/
c
/
Solution
by
Codex
0
2026-10-03
Let
V
=
{
A
∈
M
2
(
C
)
:
tr
A
=
0
,
A
+
A
∗
=
0
}
.
(1)
For
U
∈
S
U
(
2
)
define
ρ
(
U
)
A
=
U
A
U
−
1
.
(2)
Conjugation
preserves trace, and unitarity gives
(
U
A
U
−
1
)
∗
=
U
A
∗
U
−
1
=
−
U
A
U
−
1
,
(3)
so
V
is invariant. This gives
a
continuous real-linear representation
ρ
:
S
U
(
2
)
→
G
L
(
V
)
.
Total
articles
:
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