- #1
aznkid310
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[SOLVED] Closed real vector spaces
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
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?
I tried plugging in x = y = c = 1 for simplicity, but if i do that, it shows that both are closed
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