Show that the embedding of part (g) takes lines to "lines."
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please answer part h.
![{(х, у, г) € R$I(x, у, 2) # (0, 0, 0)}. Define
~ on T by (x], У1, Z1)
Ax2, y1 = Ay2,
12. Let T
(x2, y2, z2) if there exists a nonzero real number A such that x1 =
and z1 = Az2.
(a) Show that ~
is an equivalence relation on T.
(b) Give a geometric description of the equivalence class of (x, y, z).
The set T/
(x, y, z) is denoted by [x, y, z], and is called a point.
~ is called the real projective plane, and is denoted by P2. The class of
(c) Let (a, b, c) € T, and suppose that (x1, y1, Z1)
axi + byi + czi = 0, then ax2 + by2 + cz2 = 0. Conclude that
(x2, y2, z2). Show that if
L%3D ([x, у, z] € P|аx + by + cz %3D
0}
is a well-defined subset of P2. Such sets L are called lines.
(d) Show that the triples (a1, b1, c1) e T and (a2, b2, c2) E T determine the same
line if and only if (a1, b1, c1) ~ (a2, b2, c2).
(e) Given two distinct points of P?, show that there exists exactly one line that contains
both points.
(f) Given two distinct lines, show that there exists exactly one point that belongs to
both lines.
(g) Show that the function f : R2 → P² defined by f (x, y) = [x, y, 1] is a one-to-
one function. This is one possible embedding of the “affine plane" into the projective
plane. We sometimes say that P2 is the "completion" of R?.
(h) Show that the embedding of part (g) takes lines to "lines."
(i) If two lines intersect in R2, show that the image of their intersection is the inter-
section of their images (under the embedding defined in part (g)).
(j) If two lines are parallel in R², what happens to their images under the embedding
into P2?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a3c2c25-ff4e-41fc-9840-157b82142c59%2F5e8c2006-1834-4d41-a59a-c160e00ab0af%2Fph1z9gh_processed.png&w=3840&q=75)
Transcribed Image Text:{(х, у, г) € R$I(x, у, 2) # (0, 0, 0)}. Define
~ on T by (x], У1, Z1)
Ax2, y1 = Ay2,
12. Let T
(x2, y2, z2) if there exists a nonzero real number A such that x1 =
and z1 = Az2.
(a) Show that ~
is an equivalence relation on T.
(b) Give a geometric description of the equivalence class of (x, y, z).
The set T/
(x, y, z) is denoted by [x, y, z], and is called a point.
~ is called the real projective plane, and is denoted by P2. The class of
(c) Let (a, b, c) € T, and suppose that (x1, y1, Z1)
axi + byi + czi = 0, then ax2 + by2 + cz2 = 0. Conclude that
(x2, y2, z2). Show that if
L%3D ([x, у, z] € P|аx + by + cz %3D
0}
is a well-defined subset of P2. Such sets L are called lines.
(d) Show that the triples (a1, b1, c1) e T and (a2, b2, c2) E T determine the same
line if and only if (a1, b1, c1) ~ (a2, b2, c2).
(e) Given two distinct points of P?, show that there exists exactly one line that contains
both points.
(f) Given two distinct lines, show that there exists exactly one point that belongs to
both lines.
(g) Show that the function f : R2 → P² defined by f (x, y) = [x, y, 1] is a one-to-
one function. This is one possible embedding of the “affine plane" into the projective
plane. We sometimes say that P2 is the "completion" of R?.
(h) Show that the embedding of part (g) takes lines to "lines."
(i) If two lines intersect in R2, show that the image of their intersection is the inter-
section of their images (under the embedding defined in part (g)).
(j) If two lines are parallel in R², what happens to their images under the embedding
into P2?
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