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Math Help - A Couple of Vector Space Problems

  1. #1
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    A Couple of Vector Space Problems

    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)
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  2. #2
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    Quote Originally Posted by huutri030489 View Post
    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).


    A v.s. must fulfil distributivity: c\odot (x+y)=c\odot x+c\odot y ...is this true in this case?

    Tonio



    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)
    .
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