Linear Algebra Need help

[huutri030489]

1/ Prove that the set V=R+ ( the set of all positive real numbers) is a vector space with the following nonstandard operations: for any x,y belong to R+ & for any scalar c belong to R:
x O+ ( +signal into circle) y=x.y (definition of vector addition) & c O ( dot signal into circle) x = x^c (definition of scalar multiplcation) ( must verify that all 10 axioms defining a vectorspace are satisfied).
2/ Consider the vector space V = C (-infinite,infinite)= all fumctions f(x) which are continuous everywhere. Show that the following subset H of C (-infinite, infinite) is in fact a subspace of C (-infinite,infinite):
H= {all functions f(x) satisfying the differential equation f " (x) +25 f(x)=0}
(need to verify all 3 subspace requirements)

 

[DrMike]

1/ Prove that the set V=R+ ( the set of all positive real numbers) is a vector space with the following nonstandard operations: for any x,y belong to R+ & for any scalar c belong to R:
x O+ ( +signal into circle) y=x.y (definition of vector addition) & c O ( dot signal into circle) x = x^c (definition of scalar multiplcation) ( must verify that all 10 axioms defining a vectorspace are satisfied).
2/ Consider the vector space V = C (-infinite,infinite)= all fumctions f(x) which are continuous everywhere. Show that the following subset H of C (-infinite, infinite) is in fact a subspace of C (-infinite,infinite):
H= {all functions f(x) satisfying the differential equation f " (x) +25 f(x)=0}
(need to verify all 3 subspace requirements)

Ok, the 10 vector space axioms are each of the form

bla bla = yada yada.

for each one, you either
* Write down the LHS, apply your definitions of + and ., and prove it equals the RHS
* Write down the RHS, apply your definitions of + and ., and prove it equals the LHS
* Write down the LHS and the RHS, apply your definitions of + and ., and prove the expressions you get are equal.

I'll do one for you - I'll prove that c (.) (x (+) y) = c (.) x (+) c (.) y

LHS = c (.) (x (+) y) = c (.) xy = (xy)^c = x^c y^c
RHS = c (.) x (+) c (.) y = x^c (+) y^c = x^c y^c
hence LHS = RHS