MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #2
Last modified: October 3, 2007
Due Thursday, Oct. 4 at class
1.
(a) Suppose that you modify the NSA budget system from the lecture
outline by defining 0
S
=
θ
so that the axiom
∀
x,
0
x
=
θ
is satisfied.
Does this make it into a vector space, or does it create a problem with
one of the other axioms?
(b) A conjecture: In a vector space
X
, the element
z
that satisfies
z
+
z
=
z
is unique.
Either prove this from the axioms, or construct a
counterexample.
2. Prove Luenberger’s axiom 1 from the other axioms. Note that axiom 3 is
“onesided.” Without axiom 1, this is still OK, but you must prove that
θ
is a left identity if you want to use it as one.
3. Luenberger, section 2.16, problem 4. The statement that you are to prove
could have been used as the definition of “convex hull,” but Luenberger
used a different definition on page 18.
4. Find the convex hull of the Koch snowflake, a famous fractal curve, and
prove that your answer is correct. You construct the snowflake as follows.
•
Start with an equilateral triangle of side
A
.
•
In the middle third of each side, construct an equilateral triangle of
side
A
3
, pointing outwards.
•
The snowflake now consists of 12 segments of length
A
3
. In the mid
dle third of each segment, construct an equilateral triangle of side
A
9
,
pointing outwards.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Math, Linear Algebra, Vector Space, Luenberger

Click to edit the document details