M231: Exercise sheet 3
1. Here is a perspective picture of train tracks
in a desert. The sleepers on the track are
parallel and 1 metre apart. I have drawn the
rst two sleepers. Explain how to accurat
MA30231
University of Bath
DEPARTMENT OF MATHEMATICAL SCIENCES
EXAMINATION
M30231: PROJECTIVE GEOMETRY
SAMPLE EXAM, duration 2 hours
No calculators may be brought in and used.
Full marks will be given
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M231: Exercise sheet 5
1. Let v1 , . . . , vn be a basis of a vector space V and dene v1 , . . . , vn in V by
vi (vj ) =
1
0
if i = j;
otherwise.
and extending by linearity.
(a) If v =
(b) Prove
n
i=1
University of Bath
DEPARTMENT OF MATHEMATICAL SCIENCES
Outline solution to Examination Questions
Unit Code Unit Title
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M231: Exercise sheet 2
Lines and planes
1. Let A, B, C be three points in a projective plane with homogeneous coordinates
[a0 , a1 , a2 ], [b0 , b1 , b2 ] and [c0 , c1 , c2 ], respectively. Show that
M231: Exercise sheet 1
1. Let
(a)
(b)
(c)
V be a vector space.
(Very easy) Let U1 , U2 , W V with U1 , U2 W . Show that U1 + U2 W .
Deduce that if X1 , X2 , Y P(V ) with X1 , X2 Y then the join X1 X2
M231: Exercise sheet 4
On projections
1. Let L1 and L2 be distinct projective lines in a projective plane that intersect at A.
Let : L1 L2 be a projective transformation such that A = A.
Show that is