311 – HW2

January 29, 2015

Following Venema’s book, we examine a “toy” collection of incidence axioms: Our primitive (undefined) terms are point, line, and the relation “to lie on” (between a point and a line)

  1. For every pair of distinct points P and Q there exists exactly one line \ell such that both P and Q lie on \ell.
  2. For every line \ell there exist at least two distinct points P and Q such that both P and Q lie on \ell.
  3. There exist three points that do not all lie on any one line.

Axiom 3 gives us in particular that there are at least three points. However, as shown in lecture (or see example 2.2.2 in Venema’s book), we cannot prove from these axioms that there are more than three, because there is a model of these axioms with precisely three points, and three lines, each line containing exactly two of the points.

For a fixed positive integer n\ge 2, replace axiom 2 with axiom 2^n:

For every line \ell there exist at least n distinct points, all of which lie on \ell.

(In particular, axiom 2^n is the same as axiom 2.)

Now consider the theory T_n consisting of axioms 1, 2^n, and 3. The comment above indicates that the smallest possible number of points that a model of T_2 can have is three.

  1. Try to find the smallest possible number of points that a model of T_3 can have. That is, describe a model of T_3 with as few points as you can manage. (To show that the number, say k, you find is optimal, that is, it cannot be reduced, one would need to prove from the axioms of T_3 the theorem that says that there exist at least k points. That would be great, but I am not requiring that. The number k you identify does not need to be optimal, but try to make it as small as possible. Of course, for all we know at this point, it could well be that any model of T_3 is infinite.)
  2. Do the same for T_4 and T_5.
  3. If possible, can you say something in general about the number of points of the smallest model of T_n (as a function of n)?

The implicit suggestion here is to play with these theories, trying to understand what can and cannot be deduced from them. Ryan suggested to look at a variant 3^n of axiom 3, namely: Given any n points, there is another point such that the n+1 resulting points do not all lie on the same line. Is this a theorem of T_n? Are there other interesting variants we can consider?

Feel free to include in your homework any results you find about these theories or their models, even if not directly related to the three questions above.

(In principle, this set is due February 2 at the beginning of lecture, but if needed, I am fine with extending the deadline so you have time to further explore the axioms. Let me know.)

403/503 – HW5

January 29, 2015

Show that affine spaces are closed under affine combinations, that is: If C is an affine space, n is any positive integer, c_1,\dots,c_n are any vectors in C, and r_1,\dots,r_n are any reals such that


then r_1c_1+\dots+r_nc_n\in C.

(Due February 2 at the beginning of lecture.)


January 28, 2015

Last weekend was Francisco’s birthday party. (The actual birthday is at the end of the year, during the holidays.)



403/503 – HW4

January 27, 2015

Let b\in\mathbb R^m and let A be an m\times n matrix with real entries. Set C=\{x\in\mathbb R^n\mid Ax=b\}, and suppose that C\ne\emptyset. Show that C is an affine space.

(Due January 29 at the beginning of lecture.)

403/503 – HW3

January 23, 2015

Recall that a vector space V is said to be of dimension n, \dim V=n, iff there is an independent subset of V of size n, and no independent subset has size n+1.

A basis of a vector space V is any independent set whose span is V.

Suppose that V is a vector space of dimension n. In lecture we showed that a subset of V of size n is independent iff it spans V. Show the following:

  1. V has a basis, and all bases have size n.
  2. If A\subseteq V is independent, then there is a basis B of V with A\subseteq B.

(Due January 27 at the beginning of lecture.)

403/503 – HW2

January 20, 2015

Let V be a vector space. In lecture we verified that the following two statements about a set A\subset V are equivalent:

  • For any v_1,\dots,v_n\in A and any scalars r_1,\dots,r_n and s_1,\dots,s_n, if \displaystyle \sum_{i=1}^n r_i v_i=\sum_{i=1}^n s_i v_i, then r_i=s_i for all i.
  • For any v_1,\dots,v_n\in A and any scalars r_1,\dots,r_n, if \displaystyle \sum_{i=1}^n r_i v_i=0, then r_i=0 for all i.

Recall that the set A is independent iff no element of A is in the span of the other elements, that is, for any a\in A, we have that a\notin\mathrm{sp}(A\setminus\{a\}).

  1. Show that A is independent iff the two (equivalent) statements above hold.

(Due January 22 at the beginning of lecture.)

311 – HW1

January 15, 2015

Go through the first 28 propositions of Book 1 of Euclid’s Elements. Make a list of the unjustified inferences you observe in the proofs.

(Due January 20 at the beginning of lecture.)