- #1

aznkid310

- 109

- 1

**[SOLVED] Closed real vector spaces**

## Homework Statement

Determine whether the given set V is closed under the operations (+) and (.):

V is the set of all ordered pairs of real numbers (x,y) where x>0 and y>0:

(x,y)(+)(x',y') = (x+x',y+y')

and

c(.)(x,y) = (cx,cy), where c is a scalar, (.) = multiplication

## Homework Equations

To show if they are closed or not, i know that i must satisfy a set of conditions such as:

u(+)v = v(+)u

u(+)0 = u

c(.)(u+v) = c(.)u(+)c(.)(v)

et...

I also know that (x,y)(+)(x',y') = (x+x',y+y') is closed but c(.)(x,y) = (cx,cy) is not. So how do i show this? Just use arbitrary numbers?

## The Attempt at a Solution

I tried plugging in x = y = c = 1 for simplicity, but if i do that, it shows that both are closed