Question

Let -? and ? denote two distinct objects, neither of which is in R. Define an addition and scalar multiplication on R U {?} U {-?} as you could guess from the notation. Specifically, the sum and product of two real numbers is as usual, and for t ? R define


t? = { -? if t<0 , 0 if t=0, ? if t>0


t(-?) = { ? if t<0, 0 if t=0, -? if t>0


t+? = ?+t=?, t+(-?)=(-?)+t=-?, ?+?=?, (-?)+(-?)=-?, ?+(-?)=0


IsR U {?} U {-?} a vector space over R? Please Explain.

Answer 1

Answer and Step-by-step explanation:

Obviously addition is closed for R including both infinities

As t+infty = infty +t = infty and t+(-infty) =(-infty)+t =(-infty)

and inverse are -infty for infty and infty for -infty

Hence a group under addition

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But regarding multiplication

we can say 1/infty = 2/infty =0

Hence infinity*0 is not unique.

Also infinity and -infty do not have multiplicative inverse as there is no t such that t*infty = 1

Hence cannot be a vector space.