Let V be the set of all real numbers Define an operation of

Let V be the set of all real numbers. Define an operation of \"addition\" by x y = the maximum of x and y for all x, y elementof V. Define an operation of \"scalar multiplication\" by alpha x = alpha x for all alpha elementof R and x elementof V. Is this a vector space? Explain why or why not.

Solution

V is not a vector space , as the identity element does not exist wrt the given operation

x* y = max(x,y)   

Let e be te identity element then   

x* e= x where x is a real no

if x>e then x* e= x (as x is the max )

if x< e then x*e =e (as e is the max)

hence for all x identity element e doesnt exist