Linear Algebra Exercises-n-Answers.pdf

Answers to Exercises 177
4 If p 1 , p 2 , p 3 , and q 1 , q 2 , q 3 are two triples of homogeneous coordinates for p then the two column
vectors are in proportion, that is, lie on the same line through the origin. Similarly, the two row vectors
are in proportion.
⎛
k · ⎝ p ⎞ ⎛
1
p 2
⎠ = ⎝ q ⎞
1
q 2
⎠ m · (L ) ( )
1 L 2 L 3 = M1 M 2 M 3
p 3 q 3
Then multiplying gives the answer (km) · (p 1 L 1 + p 2 L 2 + p 3 L 3 ) = q 1 M 1 + q 2 M 2 + q 3 M 3 = 0.
5 The picture of the solar eclipse — unless the image plane is exactly perpendicular to the line from
the sun through the pinhole — shows the circle of the sun projecting to an image that is an ellipse.
(Another example is that in many pictures in this Topic, the circle that is the sphere’s equator is drawn
as an ellipse, that is, is seen by a viewer of the drawing as an ellipse.)
The solar eclipse picture also shows the converse. If we picture the projection as going from left to
right through the pinhole then the ellipse I projects through P to a circle S.
6 A spot on the unit sphere
⎛ ⎞
p 1
⎝p 2
⎠
p 3
is non-equatorial if and only if p 3 ≠ 0. In that case it corresponds to this point on the z = 1 plane
⎛
⎝ p ⎞
1/p 3
p 2 /p 3
⎠
1
since that is intersection of the line containing the vector and the plane.
7 (a) Other pictures are possible, but this is one.
T 0
0
V 0
V 2
U 2
T 2
T 1
U 1
V 1
The intersections T 0 U 1 ∩ T 1 U 0 = V 2 , T 0 V 1 ∩ T 1 V 0 = U 2 , and U 0 V 1 ∩ U 1 V 0 = T 2 are labeled so that
on each line is a T , a U, and a V .
(b) The lemma used in Desargue’s Theorem gives a basis B with respect to which the points have
these homogeneous coordinate vectors.
⎛ ⎞
⎛ ⎞
⎛ ⎞
⎛ ⎞
Rep B (⃗t 0 ) =
⎝ 1 0⎠ Rep B (⃗t 1 ) =
0
⎝ 0 1
0
⎠ Rep B (⃗t 2 ) =
⎝ 0 0⎠ Rep B (⃗v 0 ) =
1
⎝ 1 1
1
⎠
(c) First, any U 0 on T 0 V 0
⎛
Rep B (⃗u 0 ) = a ⎝ 1 ⎞ ⎛
0⎠ + b ⎝ 1 ⎞ ⎛
1⎠ = ⎝ a + b
⎞
b ⎠
0 1 b
has homogeneous coordinate vectors of this form ⎛
⎝ u ⎞
0
1 ⎠
1
(u 0 is a parameter; it depends on where on the T 0 V 0 line the point U 0 is, but any point on that line
has a homogeneous coordinate vector of this form for some u 0 ∈ R). Similarly, U 2 is on T 1 V 0
⎛
Rep B (⃗u 2 ) = c ⎝ 0 ⎞ ⎛
1⎠ + d ⎝ 1 ⎞ ⎛
1⎠ = ⎝
d
⎞
c + d⎠
0 1 d
and so has this homogeneous coordinate vector.
⎛
⎞
⎝ 1 u 2
⎠
1