ma253
, Fall
2007
— Problem Set
6
This is a short problem set to keep you busy until Wednesday. There are
only three problems, and all of them are starred. This assignment is due on
All Hallows Eve,
Wednesday, October 31
.
*1.
Let
P
be the linear space of all polynomials (of any degree). Show that
no finite set of polynomials
{
p
1
(
t
)
, p
2
(
t
)
, . . . , p
m
(
t
)
}
can span all of
P
.
(Hint: since there are a finite number of polynomials in my set, there is
one of largest degree.)
*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 inde
pendent?
b. What property of
A
will guarantee that Span
(
v
1
, v
2
, . . . , v
k
) =
R
n
?
c. What property of
A
will guarantee that
{
v
1
, v
2
, . . . , v
k
}
is a basis of
R
n
?
Explain your answers!
*3.
(This is closely related to problem
58
in section
4
.
1
, and to Example
1
in that section.) A
differential equation
is an equation that gives a relation
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 Fall '07
 GHITZA
 Linear Algebra, Algebra, Derivative, Hallows Eve, linear space, Short Problem

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