Math 235 Assignment 8
Due 9:15am, Wednesday March 21, 2007.
1. From the Text
§
5.4
#6. Let
T
:
P
2
→
P
4
be the transformation that maps a polynomial
p
(
t
) into the polynomial
p
(
t
) +
t
2
p
(
t
)
.
a. Find the image of
p
(
t
) = 2

t
+
t
2
.
b. Show that
T
is a linear transformation.
c. Find the matrix for
T
relative to the bases
{
1
, t, t
2
}
and
{
1
, t, t
2
, t
3
, t
4
}
.
d. Determine the rank of
T.
Is
T
injective? Is
T
surjective?
Solution.
a.
T
(2

t
+
t
2
) = 2

t
+
t
2
+
t
2
(2

t
+
t
2
) = 2

t
+ 3
t
2

t
3
+
t
4
.
b. We need to show that
T
(
a
p
1
(
t
) +
b
p
2
(
t
)) =
aT
(
p
1
(
t
)) +
bT
(
p
2
(
t
))
,
where
p
1
(
t
)
,
p
2
(
t
)
∈
P
2
and
a
and
b
are scalars.
T
(
a
p
1
(
t
) +
b
p
2
(
t
)) =
a
p
1
(
t
) +
b
p
2
(
t
) +
t
2
(
a
p
1
(
t
) +
b
p
2
(
t
))
= (
a
p
1
(
t
) +
t
2
(
a
p
1
(
t
))) + (
b
p
2
(
t
) +
t
2
(
b
p
2
(
t
)))
=
a
(
p
1
(
t
) +
t
2
p
1
(
t
)) +
b
(
p
2
(
t
) +
t
2
p
2
(
t
)) =
aT
(
p
1
(
t
)) +
bT
(
p
2
(
t
))
.
c. A general element of
P
2
is in the form
p
(
t
) =
a
+
bt
+
ct
2
,
and we
are given that
T
(
a
+
bt
+
ct
2
) =
a
+
bt
+ (
a
+
c
)
t
2
+
bt
3
+
ct
4
.
Now
coordinate vector for
p
(
t
) is
a
b
c
and coordinate vector for
T
(
p
(
t
))
is
a
b
a
+
c
b
c
.
Thus we want to find a matrix
A
such that
A
a
b
c
=
a
b
a
+
c
b
c
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
It is easy to see that
A
=
1
0
0
0
1
0
1
0
1
0
1
0
0
0
1
.
d. The rank of the matrix is 3. The transformation is injective since
a
1
b
1
c
1
=
a
2
b
2
c
2
implies
a
1
b
1
a
1
+
c
1
b
1
c
1
=
a
2
b
2
a
2
+
c
2
b
2
c
2
.
The map is not surjective.
Consider the polynomial
p
(
t
) = 1 +
t
∈
P
4
Clearly, no element of
P
2
is mapped to
p
(
t
)
.
[Optional argument:
The 3
×
3
submatrix
in the upper part of
A
is triangular, clearly has
determinant 1 and therefore has rank equal to 3. Hence rank of
A
is
at least 3. Of course its rank cannot be more than 3, the dimension
of the domain. By the rank and nullity thm, its nullity is 0. So it is
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 CELMIN
 Math, Linear Algebra, Algebra, Vector Space, minimum polynomial, xn xn+1

Click to edit the document details