Newberger Math 247 Spring 03
Homework solutions: Section 1.8 #26, 27, 31, Section 2.3 #34,36,37,38
Section 1.8 #26,27,31.
26. Let
u
and
v
be linearly independent vectors in
R
3
and let
P
be the plane
through
u
,
v
and
0
. The parametric equation of
P
is
x
=
s
u
+
t
v
(with
s
and
t
in
R
). Show that a linear transformation
T
:
R
3
→
R
3
maps
P
onto a
plane through
0
, a line through
0
or onto just the origin
0
in
R
3
. What must
be true about
T
(
u
) and
T
(
v
) in order for the image of the plane
P
to be a
plane?
We want to know the image of
P
under
T
. Any point
x
on
P
will have the
form
x
=
s
u
+
t
v
where
s
and
t
are real numbers. So
T
(
x
) =
sT
(
u
) +
tT
(
v
)
,
and
T
(
P
)
is the collection of all vectors of the form
sT
(
u
)+
tT
(
v
)
where
s
and
t
are any real number. Thus the image of
P
under
T
is Span
{
T
(
u
)
,T
(
v
)
}
.
Since there are three cases for the geometric description of the span of two
vectors in
R
3
, there are three cases for the geometric description of the image
of
P
under
T
. First
{
T
(
u
)
,T
(
v
)
}
could be independent, in which case, the
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 Spring '03
 F,newberger
 Linear Algebra, Vectors, Vector Space, linear transformation

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