Elementary Linear Algebra (Spence Insel Friedberg) Problem 11 from Chapter 7 Rev

Question

Elementary Linear Algebra (Spence Insel Friedberg)
Problem 11 from Chapter 7 Review

Determine whether the set V is a vector space with respect to the indicated operations. Justify your conclusions.

V is the set of all functions from R to R such that f(x) > 0 for all x in R. Vector addition,?, and scalar multiplication,?, are defined by:
(f?g)(x) = f(x)g(x) and (c?f)(x) = [f(x)] ^ c
for all f and g in vector V, x in R, and scalars c.

Explanation / Answer

It is vectorial space well defined , if f>0,g>0, fg>0, f^c>0 associativity (fg)h=f(gh) commutativity f*g=g*f identity element for vector addition is f(x)=1 for all x inverse element of f(x) is f^(-1)(x)=1/f(x) distributivity wrt vector addition a(f+g)=(fg)^a=f^a*g^a=af+ag distributivity wrt field addition: (a+b)f=f^(a+b)=(f^a)*f^b=af+bf distributivity wrt field multiplication (ab)f=f^(ab)=(f^b)^a=a(bf) Identity element of scalar multiplication 1f=f^1=f http://en.wikipedia.org/wiki/Vector_space#Introduction_and_definition

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