Homework assignments
Assignment 1, due Wednesday, January 15.
Long before the due date, carefully read Ziegler, Lecture 0.
Ziegler, Lecture 0: Problems 0, 5.
You don't need to do the open-ended question at the end of Problem 5.
Additional problems:
1. Show that the expression for conv(K) given on page 4 is indeed convex.
(There are two expressions for conv(K) on that page. I mean the first, more general one that does not assume a finite set.)
2.
a. Given an affine subspace S=x+L for L linear, and any point y in S, show that S=y+L.
b. Prove the the affine hull of n points is at most n-1-dimensional.
c. Prove that a set {x1,...,xn} is affinely independent if and only if there do not exist constants c1,...,cn, not all zero with Σci=0 and Σcixi=0
Assignment 2, due Wednesday, January 22.
Ziegler, Lecture 0: Problems 3, 9.
Assignment 3, due Wednesday, January 29.
Long before the due date, carefully read Ziegler, Lecture 1, Sections 1.1-1.4.
Ziegler, Lecture 1: Problems 3, 4, 5. For Problem 5: In keeping with the pattern of other Farkas Lemmas, put "but not both" into the statement.
For these problems, feel free to use anything we know from your reading in Lecture 1.
Additional Problems:
1. Fix a point x0 in a polyhedron P. Show that the lineality space of P is {y in Rd: x0+t y is in P for all t in R}.
2. Let P be a polyhedron.
If F is any nonempty face of P, then lineal(F)=lineal(P).
You'll want to use Additional Problem 1 to prove Additional Problem 2.