Projective Plane: Field Approach Consider the vector space F³ = {(x, y, z) |x, y, z = F}, where F is field. let ~ be a relation on F3 \ {0, 0, 0), defined by a = = (x1, y₁, 21) ~ B = (x2, Y2, 22) if there exists some A E K s.t. Aa=3. The projective plane over F, denoted by FP2 is the set of equivalent classes [a]. We interpret the primitive term as • point: [a], a € F³. line: {[a]la = (x, y, z), ax+by+cz = 0} for some fixed a, b, c E F, where a, b, c are not all 0. 1. Show that "" is an equivalence relation. 2. Describe the equivalent classes [a] in F3. 3. show that the line is well defined: if ax + by + cz = 0 holds for some a = (x, y, z), then it holds for any B = (x1, y₁, 21) s.t. B € [a].
Projective Plane: Field Approach Consider the vector space F³ = {(x, y, z) |x, y, z = F}, where F is field. let ~ be a relation on F3 \ {0, 0, 0), defined by a = = (x1, y₁, 21) ~ B = (x2, Y2, 22) if there exists some A E K s.t. Aa=3. The projective plane over F, denoted by FP2 is the set of equivalent classes [a]. We interpret the primitive term as • point: [a], a € F³. line: {[a]la = (x, y, z), ax+by+cz = 0} for some fixed a, b, c E F, where a, b, c are not all 0. 1. Show that "" is an equivalence relation. 2. Describe the equivalent classes [a] in F3. 3. show that the line is well defined: if ax + by + cz = 0 holds for some a = (x, y, z), then it holds for any B = (x1, y₁, 21) s.t. B € [a].
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![Projective Plane: Field Approach
Consider the vector space
F³ = {(x, y, z) |x, y, z = F},
where F is field. let ~ be a relation on F3 \ {0, 0, 0), defined by
a = = (x1, y₁, 2₁) ~ B = (x2, Y2, 22)
if there exists some A E K s.t. Aa=3. The projective plane over F, denoted
by FP2 is the set of equivalent classes [a]. We interpret the primitive term as
• point: [a], a € F³.
line: {[a]la = (x, y, z), ax+by+cz = 0} for some fixed a, b, c = F, where
a, b, c are not all 0.
1. Show that "" is an equivalence relation.
2. Describe the equivalent classes [a] in F3.
3. show that the line is well defined: if ax + by + cz = 0 holds for some
a = (x, y, z), then it holds for any B = (x1, y₁, 21) s.t. B € [a].
4. Show that the projective plane FP2 is indeed a model projective plane as
defined by axiom syatems in the previous question.
5. Show that F₂P2 is isomorphic to the Fano plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb36a358-0b1d-4556-9261-ae84b975b994%2F7bdb678f-314d-482d-be4f-a52afd744612%2Fj0043on_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Projective Plane: Field Approach
Consider the vector space
F³ = {(x, y, z) |x, y, z = F},
where F is field. let ~ be a relation on F3 \ {0, 0, 0), defined by
a = = (x1, y₁, 2₁) ~ B = (x2, Y2, 22)
if there exists some A E K s.t. Aa=3. The projective plane over F, denoted
by FP2 is the set of equivalent classes [a]. We interpret the primitive term as
• point: [a], a € F³.
line: {[a]la = (x, y, z), ax+by+cz = 0} for some fixed a, b, c = F, where
a, b, c are not all 0.
1. Show that "" is an equivalence relation.
2. Describe the equivalent classes [a] in F3.
3. show that the line is well defined: if ax + by + cz = 0 holds for some
a = (x, y, z), then it holds for any B = (x1, y₁, 21) s.t. B € [a].
4. Show that the projective plane FP2 is indeed a model projective plane as
defined by axiom syatems in the previous question.
5. Show that F₂P2 is isomorphic to the Fano plane.
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