Homework Help: Subspace, Linear Algebra, C^n[a,b]

1. Feb 25, 2010

Dustinsfl

Show that C^n[a,b] is a subspace of C[a,b] where C^n is the nth derivative.

I know the set is non empty since f(x)=x exist; however, I don't know how to start either the multiplication or addition property of subspaces to confirm that C^n is a subspace.

Thanks ahead of time for any help any of you may have.

2. Feb 25, 2010

JSuarez

What's the derivative of the sum of two functions? What's the derivative of a function multiplied by a constant?

3. Feb 25, 2010

Dustinsfl

alpha*n*f^(n-1)(x)
n*(f+g)^(n-1)(x)=n*[f^(n-1)(x)+g^(n-1)(x)]=n*f^(n-1)(x)+n*g^(n-1)(x)

That works for generalization all nth derivative functions?

4. Feb 25, 2010

JSuarez

Your expressions don't make much sense. If I'm reading them right (and please, try to use Latex), you're saying that:

$$\left(\alpha f\left(x\right)\right)^{n}=n\alpha\left(\alpha f\left(x\right)\right)^{n-1}$$

That doesn't make much sense.

5. Feb 25, 2010

Dustinsfl

multiplication: $$\alpha$$*n*$$f^{n-1}(x)$$
addition: n*$$(f+g)^{n-1}$$ = n*[$$f^{n-1}(x)$$+$$g^{n-1}(x)$$]
and then the n distributes.

Does this generalize all nth derivatives?

Is there away for me to import from Maple since I have Maple and Latex is to slow and cumbersome?

6. Feb 25, 2010

JSuarez

Those expressions are wrong, given functions f,g and a constant c, their n-th derivatives are:

$$\left(f+g\right)^{\left(n\right)}\left(x\right)=f^{\left(n\right)}\left(x\right)+g^{\left(n\right)}\left(x\right)$$

And:

$$\left(cf\right)^{\left(n\right)}\left(x\right)=cf^{\left(n\right)}\left(x\right)$$

7. Feb 25, 2010

Dustinsfl

But when the derivative is taking, don't the functions need to be multiplied by n and then the derivative is n-1?

8. Feb 25, 2010

JSuarez

You are confusing the derivative of a power with the n-th derivative.