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Math 304 Solutions 6
1 Section 4.1
17. (b) Determine the kernel and range of the following linear operator
on
R
3
.
L
(
x
) = (
x
1
,x
2
,
0)
T
Answer:
This linear transformation is the same as multiplication by
the matrix
1 0 0
0 1 0
0 0 0
.
The kernel of this linear transformation is the same as the nullspace of
the matrix. Thus, the kernel of
L
is spanned by the vector
0
0
1
.
The range of this linear transformation is the same as the column space
of the matrix. Thus, the range of
L
is spanned by the vectors
1
0
0
and
0
1
0
.
21. A linear transformation
L
:
V
→
W
is said to be
onetoone
if
L
(
v
1
) =
L
(
v
2
)
implies that
v
1
=
v
2
(i.e., no two distinct vectors
v
1
,
v
2
get mapped
into the same vector
w
∈
W
). Show that
L
is onetoone if and only if
ker(
L
) =
{
0
V
}
.
Answer:
First, we show that if
L
is onetoone, then ker(
L
) =
{
0
V
}
:
Since
L
is onetoone, there can be at most one vector that maps to
0
W
. The only vector that maps to
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 Spring '08
 STAFF

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