2.18 Invertible linear-fractional functions. Let f: R" → R" be the linear-fractional function f(x) = (Ax + b)/(c" x + d), dom f = {r|c"x+ d > 0}. %3D Suppose the matrix o- [ ] A d is nonsingular. Show that f is invertible and that f is a linear-fractional mapping. Give an explicit expression for f- and its domain in terms of A, b, c, and d. Hint. It may be easier to express f in terms of Q.
2.18 Invertible linear-fractional functions. Let f: R" → R" be the linear-fractional function f(x) = (Ax + b)/(c" x + d), dom f = {r|c"x+ d > 0}. %3D Suppose the matrix o- [ ] A d is nonsingular. Show that f is invertible and that f is a linear-fractional mapping. Give an explicit expression for f- and its domain in terms of A, b, c, and d. Hint. It may be easier to express f in terms of Q.
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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Transcribed Image Text:2.18 Invertible linear-fractional functions. Let f: R" R" be the linear-fractional function
f(x) = (Ax + b)/(c" x+d),
dom f = {r |c"x +d > 0}.
%3D
Suppose the matrix
A
Q =
d.
is nonsingular. Show that f is invertible and that f is a linear-fractional mapping.
Give an explicit expression for f and its domain in terms of A, b, c, and d. Hint. It
may be easier to express f¯' in terms of Q.
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