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# A Little Linear Algebra watch

1. (Original post by Kolya)
So can I say

, which contradicts our assumption that there exists a v such that and , therefore our sequence is a basis?
Almost, but you need to be a little more careful. You don't know that a_0 is non-zero, you just know that at least some of the a_k are non-zero.
2. (Original post by DFranklin)
Almost, but you need to be a little more careful. You don't know that a_0 is non-zero, you just know that at least some of the a_k are non-zero.
Ah right. So we just find the first a_k that is non-zero, and then multiply by to give us our non-zero coefficient next to . Thanks, Dave.
3. I'm finding the following question quite baffling: "Let be the differential operator . Prove that for a real number ."

I'm confused by this thing. It can't be just the operator acting on t, as then that just gives you . What's meant by in this context? Some kind of funky action on f(X)?

( is, I believe, the set of polynomials with real-valued coefficients.)

4. Note that if p is a polynomial, D^n(p) = 0 for all but finite n, so the sum isn't actually infinite in practice.

Spoiler:
Show
What you're being asked to prove is basically Taylor's theorem, only presented in an unusual manner.
5. Thanks for the explanation of the question. I will have a good think about it.

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Updated: November 26, 2008
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