Let V be set of all ordered pairs of real numbers with the o

Let V be set of all ordered pairs of real numbers with the operation (x, y) + (x\', y\') = (x + x\', y +y\') and r. (x, y) notequalto (0,0) Is V closed under addition? Is V closed under multiplication?

Solution

a) V is the set of ordered pairs of the form (x, y) where x, y R. If (x, y) and (x\' , y\') are two arbitrary elements of V, then ( x ,y) + (x\' , y\' ) = ( x+ x\' , y + y\' ) V as x + x\' and y + y\' belong to R. Thus V is closed under addition.

b) Since r. (x , y) = (0,0) for an arbitrary scalar r and for an arbitrary (x , y) V and since ( 0, 0) V ( as 0 is a real number and ince (0,0) is an odered pair) , therefore V is closed under scalar multiplication.However, if r. (x , y) (0,0) then r.(x,y) is not unique and then V is not closed under scalar multiplication.

Note:1. The 2nd expression defining scalar multiplication is not clear.

2. We should define this as scalar multiplication and NOT multiplication as multiplication will involve 2 elements of V.