anonymous
  • anonymous
Determine whether the following aare subspaces of C[-1,1]: (a) The set of functions f in C[-1,1] such that f(-1)=f(1) What is the procedure followed to proove these kind of questions ?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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Zarkon
  • Zarkon
I assume you know the definition of a subspace?
anonymous
  • anonymous
yes
Zarkon
  • Zarkon
what does it tell you ... that you need to show?

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anonymous
  • anonymous
closure under vector addition, scalar mult and that there is the 0 vector in it
Zarkon
  • Zarkon
ok
Zarkon
  • Zarkon
so if \(f\in C[-1,1]\) and \(f(-1)=f(-1)\) does it follow that \[cf\in C[-1,1]\]
Zarkon
  • Zarkon
\(f(-1)=f(1)\)
Zarkon
  • Zarkon
ie is \(cf(-1)=cf(1)\)
anonymous
  • anonymous
i don't really understand why they need to point out that f(-1) = f(1)
Zarkon
  • Zarkon
they are just creating a subspace with certain properties.
Zarkon
  • Zarkon
if the axioms hold then it is a subspace
Zarkon
  • Zarkon
so is \(cf(-1)=cf(1)\)?
anonymous
  • anonymous
true
Zarkon
  • Zarkon
if \(f,g\in C[-1,1]\) does it hold that \(f+g\in C[-1,1]\)?
anonymous
  • anonymous
not necessarly
Zarkon
  • Zarkon
we should also note that if \(f,g\) are continuous then so are \(cf\) and \(f+g\)
Zarkon
  • Zarkon
why?
anonymous
  • anonymous
oh ok, i understand what you meant
Zarkon
  • Zarkon
if \(f,g\in C[-1,1]\) then \(f(-1)=f(1)\) and \(g(-1)=g(1)\) thus \(f(-1)+g(-1)=f(1)+g(1)\)
Zarkon
  • Zarkon
ok?
anonymous
  • anonymous
I understand that
Zarkon
  • Zarkon
what is the zero vector?
anonymous
  • anonymous
f(0) ?
Zarkon
  • Zarkon
no...\(f(x):=0\)
Zarkon
  • Zarkon
then \(f\) is continuous and \(f(-1)=0=f(1)\) thus \(f\in C[-1,1]\)
anonymous
  • anonymous
thank, for the help but who did u know that f(−1)=0=f(1) ?
Zarkon
  • Zarkon
if \(f\) is the zero vector...ie the zero function then \(f(x)=0\) for all \(x\). That includes -1 and 1

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