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Past exam of the mathematics course of the University of Cambridge
/
2023
/
iii
/
Paper 302
/
3
/
e
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Mathematics course of the University of Cambridge
Past exam of the mathematics course of the University of Cambridge
2023
iii
Paper 302
3
2026-09-28
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Solution
e
Solution
0
0
0
e
For equal-
length
candidate
roots
, the required
inner products
are
α
1
⋅
α
2
=
−
1
,
α
2
⋅
α
3
=
−
1
,
α
1
⋅
α
3
=
0
,
∥
α
i
∥
2
=
2.
(1)
Set
A
is invalid because
α
1
⋅
α
3
=
−
1
rather than zero; indeed its three
vectors
sum
to zero and are not linearly independent.
Set
B
has all the displayed
inner products
and is linearly independent, so it is valid.
Set
C
is the standard realization
e
1
−
e
2
,
e
2
−
e
3
,
e
3
−
e
4
(2)
of the
A3 root system
and is also valid.
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Paper 302
iii
2023
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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