Homework Set # 3 SOLUTIONS Math 453
Section 6.1
19. Prove that a linear transformation is one-to-one if and only if the null space of T is cfw_0.
Solution:
Suppose T is one-to-one, and suppose there i
MATH 453: Solutions to Homework 9
10.20
Proof. Suppose the original transportation problem is
Minimize
cij xij
i,j
subject to
xij = si
j
xij = dj
i
xij 0
s
d
such that every si , dj and cij are ration
MATH 453: Solutions to Homework 7
6.19
Proof. Suppose the strong duality theorem is known. We would like to prove the
Farkas Lemma. Consider the following LP and its dual.
Minimize
0x
subject to Ax =
MATH 453: Solutions to Homework 3
2.33
Proof. We rst show that if f is concave, then its hypograph is convex. Take
two arbitrary points (x1 , y1 ) and (x2 , y2 ) from its hypograph, then by the deniti
Homework Set # 2 SOLUTIONS Math 453
Section 5.6
6. Suppose that is a norm on Rp and that A is a real nonsingular p p matrix. Dene
x A := Ax and prove that A is a norm.
Solution:
First we need to show
Homework Set # 3 SOLUTIONS Math 453
Section 7.5
11. If w is complex, show that I (2/wH w)(wwH ) is unitary.
Solution: Note that
I (2/wH w)(wwH )
H
I (2/wH w)(wwH ) = I (2/wH w)(wwH ) I (2/wH w)(wwH )
MATH 453: Solutions to Homework 5
3.25
Proof. First we rewrite the linear program into the standard form by introducing
slack variables and adding new variables for the unrestricted variables:
Minimiz
MATH 453: Solutions to Homework 8
12.1
Proof. Denote by fi,j be the amount of ow from vertex i to j. The maximal
ow is the following: f1,2 = 2, f1,3 = 4, f2,5 = 2, f3,4 = 2, f3,7 = 2, f4,5 = 1,
f4,6 =
Homework Set # 1 SOLUTIONS Math 453
Section 5.1
12. Part b - u =< 4 2 1 >T and v =< 4 0 3 >T . Find an equation of the form ax+by+cz = d
satised by the entries x, y and z in w =< x y z >T = u + v
solu