Graphics Reference
In-Depth Information
1.4.4.
Use the Gram-Schmidt algorithm to replace the vectors (1,0,1), (0,1,1), and (2,-3,-1) by
an orthonormal set of vectors that spans the same subspace.
1.4.5.
This exercise shows that equation (1.14) in Theorem 1.4.6 gives the wrong answer if the
vectors
u
i
do not form an orthonormal basis. Consider the vector
v
= (1,2,3) in
R
3
. Its
orthogonal projection onto
R
2
should clearly be (1,2,0). Let
u
1
and
u
2
be a basis for
R
2
and let
(
)
(
)
wvuu vuu
=∑
+∑
2
.
11
1
2
1
2
Ê
Ë
ˆ
¯
(a)
If
u
1
= (1,0,0) and
u
2
=
,
,
0
, show that
w
π (1,2,0).
(b)
If
u
1
= (2,0,0) and
u
2
= (0,3,0), show that
w
π (1,2,0).
Section 1.5
Suppose that
X
is a k-dimensional plane in
R
n
and that
1.5.1.
{
}
=+
{
}
,
XpvvV qwwW
=+
Œ
Œ
where
p
,
q
Œ
R
n
and
V
and
W
are k-dimensional vector subspaces of
R
n
. Show that
V
=
W
.
1.5.2.
Fill in the missing details in the proof of Proposition 1.5.2.
1.5.3.
Prove that the intersection of two planes is a plane.
1.5.4.
Prove that if
X
is a plane, then aff(
X
) =
X
.
1.5.5.
Prove Lemma 1.5.6.
1.5.6.
Prove Lemma 1.5.7.
Prove that two lines in
R
2
1.5.7.
(a)
are parallel if and only if they have parallel direction
vectors.
Let
L
and
L
¢ be lines in
R
2
defined by the equations ax + by = c and a¢x + b¢y = c¢,
respectively. Prove that
L
and
L
¢ are parallel if and only if a¢=ka and b¢=kb for
some nonzero constant k.
(b)
1.5.8.
Prove Theorem 1.5.10.
Find a basis for the plane x - 3y + 2z = 12 in
R
3
.
1.5.9.
Find the equation of all planes in
R
3
that are orthogonal to the vector (1,2,3).
1.5.10.
1.5.11.
Find the equation of the plane containing the points (1,0,1), (3,-1,1), and (0,1,1).
Find the equation for the plane in
R
3
that contains the point (1,2,1) and is parallel to
the plane defined by x - y - z = 7.
1.5.12.
Find an equation for all planes in
R
3
that contain the point (1,2,1) and are orthogonal
to the plane defined by x - y - z = 7.
1.5.13.
1.5.14.
Find an orthonormal basis for the plane x + 2y - z = 3.
1.5.15.
Let
X
be the plane defined by 2x + y - 3z = 7. Let
v
= (2,1,0).
(a)
Find the orthogonal projection of
v
on
X
.
(b)
Find the orthogonal complement of
v
with respect to
X
.
