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Given n e N and the function f : R" → R", we can define, for each nonnegative integer k, the kth iterate fk of f as follows: first let fº be the identity map on R", and then for each k > 0 let fk+1 = f o fk. If ƒ is linear, must fk necessarily be linear for all k? Prove your claim.

Given n e N and the function f : R" → R", we can define, for each nonnegative integer k, the kth iterate fk of f as follows: first let fº be the identity map on R", and then for each k > 0 let fk+1 = f o fk. If ƒ is linear, must fk necessarily be linear for all k? Prove your claim.

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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Part (f) only as soon as possible

Problem 2. Let m, n E N. Given any c e R and functions f,g : R™ → R", we can define the functions cf, f + g, and f – g by the rules (cf)(ï) = cf(ï), (f + g)(ï) = f(7) + g(ï), and (f – 9)(7) = f(F) – g(7) for all 7 e R™. (a) Let f : R" → R" be a function. Prove that if f is linear, then so is cf for every c e R. (b) Let f, g : R™ → R" be functions. Prove that if f and g are linear, then so is f +g. (c) Let f, g : R™ → R" be functions. Prove that if f and g are linear, then so is f g. (d) Let m, n, p E N. Prove that if f : R™ → R" and g : R" → RP are linear, then gof is linear. (e) Given n EN and functions f, g : R" → R, we can define the product function fg : R" → R by (fg)(F) = f(ï)g(F). If the functions f,g : R" → R are linear, must the product function fg necessarily be linear? Justify your answer. (f) Given n e N and the function f : R" → R", we can define, for each nonnegative integer k, the kth iterate fk of f as follows: first let fº be the identity map on R", and then for each k > 0 let fk+1 – f o fk. If f is linear, must fk necessarily be linear for all k? Prove your claim.
Transcribed Image Text:Problem 2. Let m, n E N. Given any c e R and functions f,g : R™ → R", we can define the functions cf, f + g, and f – g by the rules (cf)(ï) = cf(ï), (f + g)(ï) = f(7) + g(ï), and (f – 9)(7) = f(F) – g(7) for all 7 e R™. (a) Let f : R" → R" be a function. Prove that if f is linear, then so is cf for every c e R. (b) Let f, g : R™ → R" be functions. Prove that if f and g are linear, then so is f +g. (c) Let f, g : R™ → R" be functions. Prove that if f and g are linear, then so is f g. (d) Let m, n, p E N. Prove that if f : R™ → R" and g : R" → RP are linear, then gof is linear. (e) Given n EN and functions f, g : R" → R, we can define the product function fg : R" → R by (fg)(F) = f(ï)g(F). If the functions f,g : R" → R are linear, must the product function fg necessarily be linear? Justify your answer. (f) Given n e N and the function f : R" → R", we can define, for each nonnegative integer k, the kth iterate fk of f as follows: first let fº be the identity map on R", and then for each k > 0 let fk+1 – f o fk. If f is linear, must fk necessarily be linear for all k? Prove your claim.
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