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Determining if a set is a subspace

  1. Jun 21, 2009 #1
    1. The problem statement, all variables and given/known data
    Determine whether or not the set of all functions f such that f(1)+f(-1)=f(5) is a subspace of the vector space F of all functions mapping R into R.


    2. Relevant equations



    3. The attempt at a solution
    I think that
    (f(1)+f(-1))+(g(1)+g(-1))=(f+g)(1)+(f+g)(-1)=(f+g)(5)
    shows it is closed under vector addition, but I'm not sure. I'm also not sure what to do about checking scalar multiplication. I don't think that r(f(1)+f(-1)=rf(5) does anything at all for me.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 21, 2009 #2
    Your demonstration of closure under vector addition is fine. With respect to scalar multiplication, you are being asked to show that rf defined as (rf)(x) = r*f(x) is also an element of this vector space whenever f is.
     
  4. Jun 21, 2009 #3
    So (rf)(1)+(rf)(-1)=r*(f(1)+f(-1))= rf(5) which is not equal to f(5), so it's not closed under scalar multiplication?
     
  5. Jun 21, 2009 #4
    Why are we looking at f(5)? The defining axiom is that h(1) + h(-1) = h(5) for each h in the vector space.
     
  6. Jun 21, 2009 #5
    so do I only need to show (rf)(1)+(rf)(-1)=r*(f(1)+f(-1))?
     
  7. Jun 21, 2009 #6
    You also need to show that this is equivalent to (rf)(5), which may seem trivial, but necessary.
     
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