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 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

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.

mariah is studying the life cycle of the monarch butterfly. she must choose one stage in the cycle to present in science class: egg, caterpillar, chrysalis, or adult. she plans to make her choice randomly using a custom spinner divided into four sections. first, however, she spins the spinner 50 times to see the frequencies it generates.

stage times landed on

egg 10

caterpillar 12

chrysalis 15

adult 13

step-by-step explanation: