Is the periodic function a vector space?

  • Thread starter golmschenk
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  • #1
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Homework Statement


Is the following a vector space:
The set of all periodic functions of period 1? ( i.e. f(x+1)=f(x) )

Homework Equations


If v1 and v2 are in V then v1 + v2 is in V

If v1 is in V then c*v1 is in V where c is a scalar

The Attempt at a Solution


I'm thinking no because a function like:
f(x) = 2*x mod 2
If you multiply by 1/2 you have
f(x) = x mod 2

What I'm wondering is if the mod 2 is effected by the scalar multiplication of 1/2. Is it or is this not a subspace? Thanks for your time.
 

Answers and Replies

  • #2
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f(x) = 2*x mod 2
If you multiply by 1/2 you have
f(x) = x mod 2
Let's check this using [itex]x=17[/itex]:
[tex]\frac{(2\cdot5)\mod 2}{2} = \frac{\mathop{\mathrm{remainder}}(10,2)}{2} = \frac{0}{2} = 0,[/tex]​
while
[tex]5\mod 2 = \mathop{\mathrm{remainder}}(5,2) = 1.[/tex]​
Evidently, the two expressions are not the same. Can you see why this is?
 
  • #3
36
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Right. It's a scalar multiple of the entire function. Got it. Thanks!
 

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