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Definition 1. Fix a scalar product (·, ·) in R" and c E R", d e R and denote by H the open half-space consisting of points v such that (c, v) < -d. If D CH, we say that F: D → R" is a fractional linear map if there are a linear map A and a vector b E R" such that f(v) = 1 (Av+ b) (1) (c, v) + d for all v E D. Show that a fractional linear map maps lines into lines (even preserves betweenness) and that if T, is the translation by a vector p E R" that also T, o f is a fractional linear map.

Definition 1. Fix a scalar product (·, ·) in R" and c E R", d e R and denote by H the open half-space consisting of points v such that (c, v) < -d. If D CH, we say that F: D → R" is a fractional linear map if there are a linear map A and a vector b E R" such that f(v) = 1 (Av+ b) (1) (c, v) + d for all v E D. Show that a fractional linear map maps lines into lines (even preserves betweenness) and that if T, is the translation by a vector p E R" that also T, o f is a fractional linear map.

Advanced Engineering Mathematics
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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We start with a definition. Definition 1. Fix a scalar product (·, ·) in R" and c E R", d e R and denote by H the open half-space consisting of points v such that (c, v) < -d. If D C H, we say that F:D → R" is a fractional linear map if there are a linear map A and a vector b E R" such that f (v) = 1 ;(Av + b) (1) %3D (c, v) + d for all v E D. Show that a fractional linear map maps lines into lines (even preserves betweenness) and that if T, is the translation by a vector p E R" that also T, o f is a fractional linear map.
Transcribed Image Text:We start with a definition. Definition 1. Fix a scalar product (·, ·) in R" and c E R", d e R and denote by H the open half-space consisting of points v such that (c, v) < -d. If D C H, we say that F:D → R" is a fractional linear map if there are a linear map A and a vector b E R" such that f (v) = 1 ;(Av + b) (1) %3D (c, v) + d for all v E D. Show that a fractional linear map maps lines into lines (even preserves betweenness) and that if T, is the translation by a vector p E R" that also T, o f is a fractional linear map.
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