Let H ={[1 a] : a in Z} {[0 1] Define a map φ : Z → H by φ(n) = [1 n] for all n in Z . [o 1] a. Show that φ is one-to-one. (Definition of 1 − 1: If φ(x) = φ(y), then x = y. To prove this, Assume φ(x) = φ(y) and show that x = y.) b. ) Show that φ is onto. (Definition of onto: For every y ∈ H, there is x ∈ Z such that φ(x) = y. To prove this, start with a matrix in H and then find an element in Z that is mapped to that matrix.) c. Show that φ is operation preserving. (Show: φ(x+y) = φ(x)φ(y). To do this, compute φ(x+y) and then computer φ(x)φ(y) and compare them.) d Is H ≈ Z? Explain.
Let H ={[1 a] : a in Z} {[0 1] Define a map φ : Z → H by φ(n) = [1 n] for all n in Z . [o 1] a. Show that φ is one-to-one. (Definition of 1 − 1: If φ(x) = φ(y), then x = y. To prove this, Assume φ(x) = φ(y) and show that x = y.) b. ) Show that φ is onto. (Definition of onto: For every y ∈ H, there is x ∈ Z such that φ(x) = y. To prove this, start with a matrix in H and then find an element in Z that is mapped to that matrix.) c. Show that φ is operation preserving. (Show: φ(x+y) = φ(x)φ(y). To do this, compute φ(x+y) and then computer φ(x)φ(y) and compare them.) d Is H ≈ Z? Explain.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Let H ={[1 a] : a in Z}
{[0 1]
Define a map φ : Z → H by φ(n) = [1 n] for all n in Z
. [o 1]
a. Show that φ is one-to-one.
(Definition of 1 − 1: If φ(x) = φ(y), then x = y. To prove this, Assume φ(x) = φ(y) and
show that x = y.)
b. ) Show that φ is onto.
(Definition of onto: For every y ∈ H, there is x ∈ Z such that φ(x) = y. To prove this, start
with a matrix in H and then find an element in Z that is mapped to that matrix.)
c. Show that φ is operation preserving.
(Show: φ(x+y) = φ(x)φ(y). To do this, compute φ(x+y) and then computer φ(x)φ(y) and
compare them.)
d Is H ≈ Z? Explain.
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