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Determine which of the following are homomorphisms. Justify your answers. Recall that R denotes the group of real numbers with the operation of addition and that R* = R\{0} is the group of real numbers without 0 and with the operation of multiplication. ¡. Let ƒ₁ : R → R be defined by f₁(x) = -2x for all x E R. ii. Let ƒ₂ : R → R be defined by f2(x) = |x| for all x ¤ R iii. Let ƒ3 : R* → R* be defined by f3(x) = |x| for all x ER* iv. Let G be the subgroup of GL₂(R) given by G-{(61) a+0,DER} (ö |a = Let f4 GR be defined by f5 : a (81) = b.

Determine which of the following are homomorphisms. Justify your answers. Recall that R denotes the group of real numbers with the operation of addition and that R* = R\{0} is the group of real numbers without 0 and with the operation of multiplication. ¡. Let ƒ₁ : R → R be defined by f₁(x) = -2x for all x E R. ii. Let ƒ₂ : R → R be defined by f2(x) = |x| for all x ¤ R iii. Let ƒ3 : R* → R* be defined by f3(x) = |x| for all x ER* iv. Let G be the subgroup of GL₂(R) given by G-{(61) a+0,DER} (ö |a = Let f4 GR be defined by f5 : a (81) = b.

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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Determine which of the following are homomorphisms. Justify your answers. Recall that R denotes the group of real numbers with the operation of addition and that R* = R\{0} is the group of real numbers without 0 and with the operation of multiplication. i. Let f₁ : R→ R be defined by f₁(x) = -2x for all x E R. ii. Let f₂ : R → R be defined by f₂(x) = |x| for all x ER iii. Let ƒ3 : R* → R* be defined by ƒ3(x) = |x| for all ï ¤ R* iv. Let G be the subgroup of GL₂(R) given by {(61) |a +0,b=R} G = a 1 (11) - Let f4 GR be defined by f5 b.
Transcribed Image Text:Determine which of the following are homomorphisms. Justify your answers. Recall that R denotes the group of real numbers with the operation of addition and that R* = R\{0} is the group of real numbers without 0 and with the operation of multiplication. i. Let f₁ : R→ R be defined by f₁(x) = -2x for all x E R. ii. Let f₂ : R → R be defined by f₂(x) = |x| for all x ER iii. Let ƒ3 : R* → R* be defined by ƒ3(x) = |x| for all ï ¤ R* iv. Let G be the subgroup of GL₂(R) given by {(61) |a +0,b=R} G = a 1 (11) - Let f4 GR be defined by f5 b.
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