MATH 223  HOMEWORK #3
Solutions
Problem
1. 2 (a,b,c) in Section 1.5.
The set in (a) is linearly dependent because the second matrix is

2 times the
ﬁrst. In (b) there is no relation, so the set is independent. The set in (c) is again
independent.
Problem
2. 18 in Section 1.5. (You may assume that the set
S
is ﬁnite).
Suppose we have a nontrivial relation among the polynomials:
a
1
p
1
(
x
) +
.. .
+
a
n
p
n
(
x
) = 0
.
Let the polynomials be ordered so that
deg
p
1
<
deg
p
2
< ... <
deg
p
n
.
Let
m
be the largest index such that
a
m
6
= 0, and let
d
be the degree of
p
m
. The
monomial
x
d
occurs only in the polynomial
p
m
because all the other
p
i
with nonzero
a
i
have smaller degree. It follows that
a
m
x
m
= 0, hence
a
m
= 0 and we have a
contradiction.
Problem
3. 9 in Section 1.6.
Given (
a
1
,a
2
,a
3
,a
4
), we need to solve for
b
1
,b
2
,b
3
,b
4
such that
(
a
1
,a
2
,a
3
,a
4
) =
b
1
(1
,
1
,
1
,
1) +
.. .
+
b
4
(0
,
0
,
0
,
1)
.
The unique solution is
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 Spring '10
 SHALOM
 Math, Linear Algebra, Algebra, Zagreb, Pallavolo Modena, Sisley Volley Treviso, general element

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