Let C1 be the set of all differentiable functions. Show that C1 is a vector space.
Help Please
ok lets go slow, one at a time
ok
starting with number one on the list here http://tutorial.math.lamar.edu/Classes/LinAlg/VectorSpaces.aspx
first one says if \(u, v\in V\) then \(u+b\in V\) so what do we have to check in this case? we need to check that if \(f,g\) are differentiable functions, then so is \(f+g\) because that will mean it is closed under addition
but this is straightforward because we know that the derivative of a sum is the sum of the derivative, so we know that \((f+g)'=f'+g'\) and so if \(f,g\) are differentiable, then so is \(f+g\)
that takes care of axiom a from the list
ok I understand that
what does the next one say?
if it's multuple?
yes, you have to check that if \( c\) is any number, and \(f \) is a differentiable function, then \(cf\) is differentiable as well
I think it's right
because if I have f is differentiable then cf is differentiable .So, cf prime is differentiable
Yes^^
ok but I need to proof it
So: \[(cf)' = c(f)' \] Which is definitely a differentiable function.
I want to proof the question not this
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