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Let ∞ and -∞ be two objects, not in R. We will define an addition/scalar multiplication on the set Ru{∞}u{-} by t∞ = t(-∞): t + ∞ = ∞ + t = ∞ t + (-∞) = − ∞ + t = − ∞ ∞ + ∞ = ∞ -∞ + (-∞) = = -∞ ∞ + (-∞) = 0 -∞ 0 ∞ = 0 -∞ t < 0 t = 0 t> 0 t<0 t = 0 t> 0 (1) (2) (3) (4) (5) (6) (7) (and the sum and product of real numbers is as usual). Is RU {∞} U{-} a vector space over R? If so, prove it (ie, verify all the axioms); if not, show explicitly which axiom fails.

Let ∞ and -∞ be two objects, not in R. We will define an addition/scalar multiplication on the set Ru{∞}u{-} by t∞ = t(-∞): t + ∞ = ∞ + t = ∞ t + (-∞) = − ∞ + t = − ∞ ∞ + ∞ = ∞ -∞ + (-∞) = = -∞ ∞ + (-∞) = 0 -∞ 0 ∞ = 0 -∞ t < 0 t = 0 t> 0 t<0 t = 0 t> 0 (1) (2) (3) (4) (5) (6) (7) (and the sum and product of real numbers is as usual). Is RU {∞} U{-} a vector space over R? If so, prove it (ie, verify all the axioms); if not, show explicitly which axiom fails.

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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Let ∞and -∞o be two objects, not in R. We will define an addition/scalar multiplication on the set Ru{o}u{-∞o} by too = t + ∞ = ∞ + t = ∞ t + (− ∞) = − ∞ + t = − ∞ ∞ + ∞ = ∞ -∞ + (-∞) = − ∞ ∞ + (-∞ ) = 0 ·∞ 0 ∞ ∞ t(-∞) = {0 -∞ t < 0 t = 0 t> 0 t<0 t = 0 t> 0 (1) (2) (3) (4) (5) (6) (7) (and the sum and product of real numbers is as usual). Is RU {∞} U{-∞} a vector space over R? If so, prove it (ie, verify all the axioms); if not, show explicitly which axiom fails.
Transcribed Image Text:Let ∞and -∞o be two objects, not in R. We will define an addition/scalar multiplication on the set Ru{o}u{-∞o} by too = t + ∞ = ∞ + t = ∞ t + (− ∞) = − ∞ + t = − ∞ ∞ + ∞ = ∞ -∞ + (-∞) = − ∞ ∞ + (-∞ ) = 0 ·∞ 0 ∞ ∞ t(-∞) = {0 -∞ t < 0 t = 0 t> 0 t<0 t = 0 t> 0 (1) (2) (3) (4) (5) (6) (7) (and the sum and product of real numbers is as usual). Is RU {∞} U{-∞} a vector space over R? If so, prove it (ie, verify all the axioms); if not, show explicitly which axiom fails.
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