MA 35100 HOMEWORK ASSIGNMENT #11 SOLUTIONS
Problem 1.
pg. 218; prob. 28
Consider an orthogonal transformation
L
from
R
n
to
R
n
. Show that
L
preserves the dot product:
~v
·
~w
=
L
(
~v
)
·
L
(
~w
)
,
for all
~v
and
~w
in
R
n
.
Solution:
The idea is to express the dot product in terms of lengths of vectors. For any
~x
and
~
y
in
R
n
, we have

~x
+
~
y

2
= (
~x
+
~
y
)
·
(
~x
+
~
y
) =

~x

2
+ 2 (
~x
·
~
y
) +

~
y

2
.
We may write the dot product as
~x
·
~
y
=
1
2

~x
+
~
y

2
 
~x

2
 
~
y

2
.
Let
~x
=
L
(
~v
) and
~
y
=
L
(
~w
). Then we have
L
(
~v
)
·
L
(
~w
) =
1
2

L
(
~v
) +
L
(
~w
)

2
 
L
(
~v
)

2
 
L
(
~w
)

2
=
1
2

L
(
~v
+
~w
)

2
 
L
(
~v
)

2
 
L
(
~w
)

2
(
L
is linear)
=
1
2

~v
+
~w

2
 
~v

2
 
~w

2
(
L
preserves lengths)
=
~v
·
~w.
Problem 2.
pg. 218; prob. 29
Show that an orthogonal transformation
L
from
R
n
to
R
n
preserves angles: The angle between
two nonzero vectors
~v
and
~w
in
R
n
equals the angle between
L
(
~v
) and
L
(
~w
). Conversely, is any
linear transformation that preserves angles orthogonal?
Solution:
Let
θ
be the angle between
~v
and
~w
, and
ϑ
be the angle between
L
(
~v
) and
L
(
~w
). We
want to show
ϑ
=
θ
. According to Definition 5.1.12 on page 196 of the text, we have
cos
θ
=
~v
·
~w

~v
 
~w

and
cos
ϑ
=
L
(
~v
)
·
L
(
~w
)

L
(
~v
)
 
L
(
~w
)

.
Since
L
is an orthogonal transformation, we know that

L
(
~v
)

=

~v

and

L
(
~w
)

=

~w

.
We
showed in Exercise 5.3.28 (i.e., Problem 1 above) that
L
(
~v
)
·
L
(
~w
) =
~v
·
~w
. This gives cos
ϑ
= cos
θ
,
so
ϑ
=
θ
as desired.
Conversely,
a linear transformation that preserves angles is not orthogonal.
As an example, con
sider the linear transformation
L
:
R
2
→
R
2
defined by
L
(
~x
) = 2
~x
. Then

L
(
~x
)

= 2

~x

so that
L
is not orthogonal, but
L
(
~v
)
·
L
(
~w
)

L
(
~v
)
 
L
(
~w
)

=
2
~v
·
2
~w
2

~v
 ·
2

~w

=
~v
·
~w

~v
 
~w

1
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so that
L
does indeed preserve angles.
Remark:
A linear transformation
L
that preserves the
dot product
preserves both angles and
lengths – and hence must be orthogonal. Here’s why. Let
~u
i
=
L
(
~e
i
) be the image of the standard
basis vector for
i
= 1
,
2
, . . . , n
, so that the matrix of
L
is
A
=
~u
1
~u
2
· · ·
~u
n
. Then
~u
i
·
~u
j
=
L
(
~e
i
)
·
L
(
~e
j
) =
~e
i
·
~e
j
=
(
1
if
i
=
j
,
0
otherwise.
Hence the columns of
A
form an orthonormal basis of
R
n
, so that
L
is indeed an orthogonal
transformation. Combining this with Exercise 5.3.28 (i.e., Problem #1 above) we conclude that a
linear transformation
L
:
R
n
→
R
n
is orthogonal if and only if
L
preserves the dot product.
Problem 3.
pg. 218; prob. 31
Are the
rows
of an orthogonal matrix
A
necessarily orthonormal?
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 Spring '08
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 Linear Algebra, Algebra, Dot Product, linear transformation

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