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Riemann Integrability, Linear Transformations

  • Thread starter missavvy
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  • #1
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Homework Statement



If f,g are Riemann integrable on [a,b], then for c,d real numbers,

(let I denote the integral from a to b)

I (cf + dg) = c I (f) + d I (g)


Homework Equations





The Attempt at a Solution



I have the proofs for

c I(f) = I (cf)

and

I (f+g) = I (f) + I(g)

I'm wondering how would I put them together to prove the statement?
Can I just say that because I know these 2 propositions are true, then any linear combination would also work?
 

Answers and Replies

  • #2
nicksauce
Science Advisor
Homework Helper
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Should be easy to put them together.

I(cf + cg) = I(cf) + I(dg) (since you know I(f+g) = I(f) + I(g), just replace f with cf and g with dg)
= cI(f) + dI(g) (since you know I(cf) = I(f))

I think the only problem with this is that you have to prove that if f is Riemann integrable then so is cf, which shouldn't be hard.
 

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