ma253
,Fa
l
l
2007
— Problem Set
6
Solutions
*1.
Let
P
be the linear space of all polynomials (of any degree). Show that no ±nite
set of polynomials
{
p
1
(
t
)
,p
2
(
t
)
,...,p
m
(
t
)
}
canspanallof
P
.
(Hint: since there are a ±nite number of polynomials in my set, there is one of
largest degree.)
Let
N
be the largest of the degrees of the polynomials
p
1
(
t
)
,p
2
(
t
)
,...,p
m
(
t
)
.Then
no linear combination of these polynomials can have degree greater than
N
.S
i
n
c
e
P
does contain polynomials of degree greater that
N
(e.g., it contains
t
N
+
1
), the set
{
p
1
(
t
)
,p
2
(
t
)
,...,p
m
(
t
)
}
does not span all of
P
.
*2.
Suppose we are given a set of vectors
{
~
v
1
,
~
v
2
,...,
~
v
k
}
in
R
n
. Then we can create
an
n
×
k
matrix by using the vectors
~
v
i
, in order, as the columns. Call that matrix
A
.
a. What property of
A
will guarantee that
{
~
v
1
,
~
v
2
,...,
~
v
k
}
is linearly indepen-
dent?
Independence tells us that the equation
x
1
~
v
1
+
x
2
~
v
2
+
···
+
x
k
~
v
k
=
~
0
has a unique solution, namely setting all the
x
i
to zero.
Now, given how we deFned the matrix
A
, this is the same as saying that the equa-
tion
A
~
x
=
~
0
has a unique solution (namely,
~
x
=
~
0
). That will happen if and only if