Hey there! Sign in to join this conversationNew here? Join for free
x Turn on thread page Beta
    Offline

    17
    ReputationRep:
    (Original post by Kolya)
    So can I say
    \sum_0 ^{n-1} a_k T^k(v) = 0 \Rightarrow T^{n-1} (\sum_0 ^{n-1} a_k T^k(v)) = T^{n-1}(0) = 0
    \Rightarrow \alpha _0 T^{n-1}(v) = 0 , which contradicts our assumption that there exists a v such that T^{n-1}(v) \neq 0 and a_0 \neq 0, therefore our sequence v, T(v),...,T^{n-1}(v) 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.
    • Thread Starter
    Offline

    14
    (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 T^{n-1-k} to give us our non-zero coefficient next to T^{n-1}(v). Thanks, Dave.
    • Thread Starter
    Offline

    14
    I'm finding the following question quite baffling: "Let D: \mathbb{R}[X] \to \mathbb{R}[X] be the differential operator D(f(X)) = f'(X). Prove that e^{Dt}f(X) = f(X + t) for a real number t \in \mathbb{R}."

    I'm confused by this e^{Dt} thing. It can't be just the operator acting on t, as then that just gives you e^0 = 1. What's meant by e^{Dt} in this context? Some kind of funky action on f(X)?

    (\mathbb{R}[X] is, I believe, the set of polynomials with real-valued coefficients.)
    Offline

    17
    ReputationRep:
    \displaystyle e^{Dt} = \sum_0^\infty \frac{t^n D^n}{n!}

    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.
    • Thread Starter
    Offline

    14
    Thanks for the explanation of the question. I will have a good think about it.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: November 26, 2008
Poll
Do I go to The Streets tomorrow night?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.