Mathematics 138
Problem Set 1 due at class on Wednesday, Feb. 16, 2005
It should be possible to do all these problems after the lecture on February 9.
Recommended reading:
Ryan, Chapter 0, the first two pages of Chapter 1, and page 189.
Yaglom, Introduction, skipping over the details.
You might also want to read Bennett, Chapter 3 (on reserve in Cabot), skipping anything that refers to projective geometry.
Ryan (p.8) introduces vectors as pairs of real numbers and states their properties (provable but not proved) in Theorem 1.  Yaglom (p. 243) states these same properties as axioms and indicates in a footnote that the axioms are not independent.
Here are two independent axioms for vector addition, based on Hilbert's axioms for arithmetic:

A1. Vector addition is associative: (a + b) + c = a + (b + c).
A2. Given two vectors a and b, there is a unique vector x such that a + x = b
and a unique vector y such that y + a = b.
Note that nothing has been said about zero, inverses, or commutativity, and that subtraction has not been defined!

1. Suppose that a + x = a and y + a = a.  Prove that x = y.  One way to proceed is to start with a + a = a + a and manipulate it into the form x = y.  You will need to use the "uniqueness" part of A2 as well as the "existence" part. This lets you conclude (generically) that if p + q = p + r = s, then q = r. 

2. Now suppose, in accordance with the preceding problem, that a + x = x + a = a and that
b + z = z + b = b.
Prove that x = z.  This establishes, as a theorem, the existence of a zero vector 0 such that 0 + a = a + 0 = a for any vector a.

3. Finally, suppose that a + p = 0 and that q + a = 0.  Prove that p = q, even though you have not assumed in general that a + b = b + a.

4. By considering (a + b)(1 + 1) and using distributive laws from group II on p.243 of Yaglom or parts vi. and vii. of Ryan's Theorem 1, prove that vector addition is commutative, i.e. that
a + b = b + a for any vectors a and b.
5. Prove Desargues' theorem (the "concurrent" case) as a theorem of Euclidean geometry by using vectors. The proof will be similar in style to the proof of Pappus' theorem done in lecture.

Use the data for the "medium affine faculty senate." Choose the planning committee, with Kate as the additive identity and Dave as the multiplicative identity.
6. Add Emma and Neil, first using Adam as the auxiliary point, then using Phil. Draw a single diagram representing the addition, attaching a committee name to each line and an instructor name to each point. Identify the the two "triangles" that are related by Desargues' theorem.
7. Multiply Emma and Neil, then multiply Neil by Emma, in each case using Adam as the auxiliary point. Draw a single diagram representing the multiplication in both orders, attaching a committee name to each line and an instructor name to each point. Identify the the degenerate hexagon of six instructors (three instructors on each of two intersecting committees) to which Pappus' Theorem must be applied to show that the multiplication is commutative. Don't be alarmed that your diagram has two distinct Euclidean points labeled
"Dave."
8. In the "large affine faculty senate" two "triangles" are Patty-Gavin-James (ABC) and Larry-Sally-Terry (A'B'C').  Show that these two triangles satisfy the assumptions for the converse of Desargues' Theorem and verify that the conclusion also holds.  Draw a diagram showing all the instructors and committees that are involved. Using the software on the Web site to find intersections of committees speeds thing up.

9. Choose the union committee in the large affine faculty senate, with
Wally as the additive identity O, Isaac as A, Alice as C, Larry as E.
Choose James as auxiliary instructor B.
Carry out the addition (A+C)+E and A+(C+E), using auxiliary instructors B' and D exactly as described in the notes and in the diagrams on the Web.  
a. Who is B', and who is D?
b. Who is the sum A+C+E?
c. Redraw the diagram from the Web that shows that addition is associative, with names attached to all the instructors and committees.  If two distinct points in the diagram have the same name, don't worry -- the diagram is for Euclidean geometry and you are doing finite geometry.

In 1903 Hessenberg proved that Desargues' Theorem in the plane is a consequence of Pappus' Theorem (which he called Pascal's Theorem).  Given that Pappus' Theorem was known in antiquity and so was Desargues' Theorem (though not by that name) it is remarkable how long it took for this proof to be invented. Here is the diagram that Hessenberg used:

AB is parallel to A'B' and is extended to N
A'C' is parallel to AC and is extended to L
These are the two parallel pairs of sides needed for Desargues' Theorem
AA', BB', and CC' meet at O, as required by Desargues' Theorem.
The task is to prove that BC and B'C' are parallel, which will establish Desargues' Theorem.
Constructed lines:
LM passes through A and is parallel to BB'. It intersects A'C' at L
LB' intersects AB at N
N is joined to O and also to M.
You may assume that Pappus' theorem holds even if the original two lines fail to intersect.
10. Using Pappus' theorem, prove that ON is parallel to A'L (and hence also to A'C' and AC'). Specify the six points, three on each of two non-parallel lines, that you are using
11. Using Pappus' theorem, prove that MN is parallel to BC.  Then use it again to show that MN is parallel to B'C'.  How does this complete the proof?