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## frank0520 one year ago Determine if the set is a vector space subspace. If the set is not a subspace identify which of the axioms fails.

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1. frank0520

2. anonymous

there are three things you need to check is the zero vector there, so for example in question a) does the zero function have x intercepts at multiples of $$\pi$$ and the answer is yes, since it is identically zero

3. anonymous

second is the sum of two vectors there i.e. if $$f,g\in C^{\infty}$$ with zeros at integer multiples of $$\pi$$ does $$f+g$$ have zeros at integer multiples of $$\pi$$ again the answer is yes, since by definition $$(f+g)(x)=f(x)+g(x)$$ so if $$f,g$$ is zero at multiples of $$\pi$$ does does $$f+g$$

4. anonymous

finally are scalar multiples there, i.e. if $$f$$ has zeros at integer multiples of $$\pi$$ does $$cf$$ have zeros at integer multiples of $$\pi$$ and again the answer is yes, since multiplying a function by a constant does not change the zeros

5. anonymous

repeat for each b) is zero an odd function? is the sum of two odd functions odd? is a constant multiple of an odd function odd? etc

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