Graphics Reference
In-Depth Information
1.5.16.
Find the point-normals equation for the line
x
=-
=
=+
13
2
3
t
yt
z
t
1.5.17.
Determine whether the halfplanes 2y - x ≥ 0, y - 2x + 2 ≥ 0, and -4y + 2x + 4 £ 0 have
a nonempty intersection or not.
Let V and W be subspaces of R n of dimension s and t, respectively. Assume that s + t
≥ n. Prove that V is transverse to W if and only if one of the following holds:
(a)
1.5.18.
V + W = R n .
(b)
If v 1 , v 2 ,..., v s and w 1 , w 2 ,..., w t are bases for V and W , respectively, then the
vectors v 1 , v 2 ,..., v s , w 1 , w 2 ,..., w t span R n .
Definition. Any subset X in C n of the form
1.5.19.
{
}
Xp v v
=+ +
t
t
++
...
t
v
t
,
t
,...,
t
Œ
C
,
11
2 2
kk
1 2
k
where p is a fixed point and the v 1 , v 2 ,..., v k are fixed linearly independent vectors in
C n is called a complex k-dimensional plane (through p ). If k = 1, then X is called a
complex line .
Prove that a complex line in C 2 can also be expressed as the set of points (x,y) Œ
C 2 satisfying an equation of the form
(a)
ax
+=,
by
c
for fixed a, b, c Œ C with (a,b) π (0,0).
Prove that the real points of a complex plane in C n lie on a plane in R n .
(b)
Section 1.6
1.6.1.
Prove Lemma 1.6.1.
Determine whether the following pairs of ordered bases of R 2
1.6.2.
determine the same
orientation:
(a) ((1,-2), (-3,2)) and ((1,0), (-2,3))
(b) ((-1,1), (1,2)) and ((1,-2), (1,-4))
Solve this exercise in two ways: First, use only the definition of orientation and then
check your answer using the matrix approach of Lemma 1.6.4.
1.6.3.
Why is “Does ((1,-2), (-2,4)) induce the standard orientation of the plane?” a mean-
ingless question?
1.6.4.
(a)
Find a vector (a,b) so that the basis ((-2,-3), (a,b)) determines the standard orien-
tation of the plane.
(b)
Find a vector (a,b,c) so that the basis ((2,-1,0), (-2,-1,0), (a,b,c)) determines the
standard orientation of 3-space.
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