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Consider R = {(ro, ₁,...) : x₁ = R} the vector space of countably infinite sequences of reals. We define the following maps: L(xo, 1, 2,...) = (x₁,₂,...) we call L the left shift transformation and we call R the right shift transformation. 1. Prove that L is onto but not one-to-one. 2. Prove that R is one-to-one but not onto. 3. Why are these transformations special? Could such maps exist for a finite dimensional space? R(xo, 1, 2,...) = (0, 0, 1, 2,...)

Consider R = {(ro, ₁,...) : x₁ = R} the vector space of countably infinite sequences of reals. We define the following maps: L(xo, 1, 2,...) = (x₁,₂,...) we call L the left shift transformation and we call R the right shift transformation. 1. Prove that L is onto but not one-to-one. 2. Prove that R is one-to-one but not onto. 3. Why are these transformations special? Could such maps exist for a finite dimensional space? R(xo, 1, 2,...) = (0, 0, 1, 2,...)

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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Consider R{(x0, 1,...): x₁ = R} the vector space of countably infinite sequences of reals. We define the following maps: L(xo, 1, 2,...) = (x₁,x₂,...) R(x0, x1,x2,...) = (0, 0, 1, 2, ...) we call L the left shift transformation and we call R the right shift transformation. 1. Prove that L is onto but not one-to-one. 2. Prove that R is one-to-one but not onto. 3. Why are these transformations special? Could such maps exist for a finite dimensional space?
Transcribed Image Text:Consider R{(x0, 1,...): x₁ = R} the vector space of countably infinite sequences of reals. We define the following maps: L(xo, 1, 2,...) = (x₁,x₂,...) R(x0, x1,x2,...) = (0, 0, 1, 2, ...) we call L the left shift transformation and we call R the right shift transformation. 1. Prove that L is onto but not one-to-one. 2. Prove that R is one-to-one but not onto. 3. Why are these transformations special? Could such maps exist for a finite dimensional space?
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