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Homework 1 Foundations of Computational Math 1 Fall
2010
The solutions will be posted on Wednesday, 9/8/09
Problem 1.1
This problem considers three basic vector norms:
k
.
k
1
,
k
.
k
2
,
k
.
k
∞
.
1.1.a
. Prove that
k
.
k
1
is a vector norm.
1.1.b
. Prove that
k
.
k
∞
is a vector norm.
1.1.c.
Consider
k
.
k
2
.
(i) Show that
k
.
k
2
is deﬁnite.
(ii) Show that
k
.
k
2
is homogeneous.
(iii) Show that for
k
.
k
2
the triangle inequality follows from the Cauchy inequality

x
H
y
 ≤ k
x
k
2
k
y
k
2
.
(iv) Assume you have two vectors
x
and
y
such that
k
x
k
2
=
k
y
k
2
= 1 and
x
H
y
=

x
H
y

, prove the Cauchy inequality holds for
x
and
y
.
(v) Assume you have two arbitrary vectors ˜
x
and ˜
y
. Show that there exists
x
and
y
that satisfy the conditions of part (iv ) and ˜
x
=
αx
and ˜
y
=
βy
where
α
and
β
are scalars.
(vi) Show the Cauchy inequality holds for two arbitrary vectors ˜
x
and ˜
y
.
Problem 1.2
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This note was uploaded on 07/25/2011 for the course MAD 5403 taught by Professor Gallivan during the Spring '11 term at University of Florida.
 Spring '11
 Gallivan

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