Question

Let C1 be the set of all differentiable functions. Show that C1 is a vector space.

Answer 1

Help Please

Answer 2

ok lets go slow, one at a time

Answer 3

ok

Answer 4

starting with number one on the list here
http://tutorial.math.lamar.edu/Classes/LinAlg/VectorSpaces.aspx

Answer 5

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

Answer 6

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 \(f+g\)

Answer 7

that takes care of axiom a from the list

Answer 8

ok I understand that

Answer 9

what does the next one say?

Answer 10

if it's multuple?

Answer 11

yes, you have to check that if \( c\) is any number, and \(f \) is a differentiable function, then \(cf\) is differentiable as well

Answer 12

I think it's right

Answer 13

because if I have f is differentiable then cf is differentiable .So, cf prime is differentiable

Answer 14

Yes^^

Answer 15

ok but I need to proof it

Answer 16

So:
\[(cf)' = c(f)' \]
Which is definitely a differentiable function.

Answer 17

I want to proof the question not this