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Derivative as Linear Transformation

I've attached the problem statement. I was thinking of defining the derivative, L(h) as some differential df_{t}*[f_{1}(t) f_{2}(t) ... f_{m}(t)] with df_{t}(h)=f'(t)*h. I just need to check that it satisfies the definition of lim h->^{f(t+h)-f(t)-L}^{(h)}/h but I'm getting stuck. Is the the right approach? Any suggestions?

Re: Derivative as Linear Transformation

Could you give an example, a little more info on how this was explained to you?

I think you're talking about polynomial spaces in the "power" basis, where the basis is {1, x, x^2, ... }.

You will have a vector that represents a polynomial in the "power" basis (other bases do exist, of course, and some may actually be better than the power basis for solving this problem, but I assume that's what you're using), and your idea is to multiply it by a matrix and produce another vector that will represent the derivative of the original polynomial.

If that's not what you're trying to do, let me know.

Re: Derivative as Linear Transformation

It was explained in the context of a Real Analysis class. We haven't really haven't talked about bases, so I'm not sure if that is the way to solve this. This way I have was the only thing that made some sense to me, but I'm having issues articulating it in a proof, so I thought I might be wrong.

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Re: Derivative as Linear Transformation

See the pdf. I think this is more or less what your teacher is asking for ... I'd think you would've encountered this before Real Analysis, though.

I'm going to add another pdf too, also made by me. It might be that you want to use differentials.

I hope this helps. I may be missing the point.

Re: Derivative as Linear Transformation

I'm sorry, this really isn't making sense. Thanks for trying though.