Linear Algebra

346 Chapter 4. Determinants
[Ryan], and [Eggar]). But, note that the other possible approach, the synthetic
approach of deriving the results from an axiom system, is both extraordinarily
beautiful and is also the historical route of development. Two fine sources for
this approach are [Coxeter] or[Seidenberg]. An interesting, easy, application is
[Davies]
Exercises
1 What is the equation of this point?
�1�
2 (a) Find the line incident on these points in the projective plane.
� � � �
1 4
2 , 5
3 6
(b) Find the point incident on both of these projective lines.
� 1 2 3 � , � 4 5 6 �
0
0
3 Find the formula for the line incident on two projective points. Find the formula
for the point incident on two projective lines.
4 Prove that the definition of incidence is independent of the choice of the representatives
of p and L. That is, if p1, p2, p3, andq1, q2, q3 are two triples of
homogeneous coordinates for p, andL1, L2, L3, andM1, M2, M3 are two triples
of homogeneous coordinates for L, provethatp1L1 + p2L2 + p3L3 =0ifandonly
if q1M1 + q2M2 + q3M3 =0.
5 Give a drawing to show that central projection does not preserve circles, that a
circle may project to an ellipse. Can a (non-circular) ellipse project to a circle?
6 Give the formula for the correspondence between the non-equatorial part of the
antipodal modal of the projective plane, and the plane z =1.
7 (Pappus’s Theorem) Assume that T0, U0, andV0 are collinear and that T1, U1,
and V1 are collinear. Consider these three points: (i) the intersection V2 of the lines
T0U1 and T1U0, (ii) the intersection U2 of the lines T0V1 and T1V0, and (iii) the
intersection T2 of U0V1 and U1V0.
(a) Draw a (Euclidean) picture.
(b) Apply the lemma used in Desargue’s Theorem to get simple homogeneous
coordinate vectors for the T ’s and V0.
(c) Find the resulting homogeneous coordinate vectors for U’s (these must each
involve a parameter as, e.g., U0 couldbeanywhereontheT0V0 line).
(d) Find the resulting homogeneous coordinate vectors for V1. (Hint: it involves
two parameters.)
(e) Find the resulting homogeneous coordinate vectors for V2. (It also involves
two parameters.)
(f) Show that the product of the three parameters is 1.
(g) Verify that V2 is on the T2U2 line..