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.

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