Newberger Math 247 Spring 03
Homework solutions: Section 4.1 #58
In Exercises 58 determine if the given set is a subspace of
P
n
for an
appropriate value of
n
. Justify your answer.
5. All polynomials of the form
p
(
t
) =
at
2
.
The set of all polynomials of the form
p
(
t
) =
at
2
is all scalar multi
ples of the polynomial
t
2
, thus this set is just Span
{
t
2
}
. Since all spans
are subspaces, this set is a subspace.
6. All polynomials of the form
p
(
t
) =
a
+
t
2
.
Let
H
denote the set of all polynomials of the form
p
(
t
) =
a
+
t
2
.
Then we can check that the zero polynomial
0 + 0
t
+ 0
t
2
is not in
H
.
We try to find a number
a
such that
a
+
t
2
= 0 + 0
t
+ 0
t
2
. Since the
coefficient on
t
2
on the LHS is 1, but the coefficient on
t
2
on the RHS is
0, these can never be equal (since
0
6
= 1
). Thus the zero polynomial is
not in
H
, since there is no value of
a
that will make
a
+
t
2
= 0+0
t
+0
t
2
.
Since
H
does not contain the zero polynomial,
H
is not a subspace.
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 Spring '03
 F,newberger
 Polynomials, Complex number

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