pp. 73
PExercise 1.
Which of the following maps
T
from
R
2
into
R
2
are linear
transformations?
(a)
T
(
x
1
,x
2
) = (1 +
x
1
,x
2
)
; No, because
T
(0
,
0)
6
= (0
,
0)
.
(b)
T
(
x
1
,x
2
) = (
x
2
,x
1
);
Yes, because
x
2
and
x
1
are linear
homogeneous functions of
x
1
,x
2
.
(c)
T
(
x
1
,x
2
) = (
x
2
1
,x
2
)
;
No, because, say,
2
T
(1
,
0)
6
=
T
(2
,
0)
.
(d)
T
(
x
1
,x
2
) = (sin
x
1
,x
2
);
No, since
2
T
(
π
2
,
0) = (2
,
0)
6
= (0
,
0) =
T
(
π,
0)
.
(e)
T
(
x
1
,x
2
) = (
x
1

x
2
,
0)
. Yes, because
x
1

x
2
and
0
are linear
homogeneous functions of
x
1
,x
2
.
Exercise 3.
Find the range, rank, null space, and nullity for th e differen
tiation transformation
D
on the space of polynomials of degree
≤
k
:
D
(
f
) =
f
0
.
Do the same for the integration transformation
T
:
T
(
f
) =
Z
x
0
f
(
t
)
dt.
Solution:
The range of
D
consists of all polynomials of degree strictly less
than
k
, since any polynomial
p
(
x
) =
a
n
x
n
+
...
+
a
0
is the derivative of the polynomial
R
p
(
x
) =
a
n
n
+1
x
n
+1
+
...
+
a
0
x
. The null space of
D
consists of all constants. Hence,
the rank of
D
is
k
, and the nullity is
1
.
The range of
T
consists of all continuous functions
f
such that
f
has
con
tinuous
ﬁrst derivative and
f
(0) = 0
. The null space of
T
is trivial, because if a
function is not identically zero then so is its integral. Hence, the rank of
T
is
inﬁnite, and the nullity is
0
.
Exercise 7.
Let
F
be a subﬁeld of the complex numbers and let
T
be the
function from
F
3
into
F
3
deﬁned by
T
(
x
1
,x
2
,x
3
) = (
x
1

x
2
+ 2
x
3
,
2
x
1
+
x
2
,

x
1

2
x
2
+ 2
x
3
)
(a) Verify that
T
is a linear transformation.
(b) If
(
a,b,c
)
is a vector in
F
3
, what are the conditions on
a
,
b
and
c
that the
vector be in the range of
T
? What is the rank of
T
?
(c) What are the conditions on