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Linearly independent sets within repeated powers of a linear operator

  1. Aug 6, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose that T:W -> W is a linear transformation such that Tm+1 = 0 but Tm ≠ 0. Suppose that {w1, ... , wp} is basis for Tm(W) and Tm(uk) = wk, for 1 ≤ k ≤ p. Prove that {Ti(uk) : 0 ≤ i ≤ m, 1 ≤ j ≤ p} is a linearly independent set.


    2. Relevant equations



    3. The attempt at a solution
    By definition of a basis w1, ..., wp are linearly independent. Now suppose Tm - 1(uk) is not linearly independent from the w's. Then it can be written as some sum of linear combinations of the w's which is equivalent to saying Tm-1(uk) = c1Tmu1 + ... + cpTmup. If both sides are left multiplied by T then we have Tm = the linear combination of Tm + 1 of the ui's which by definition are all 0. But then we have Tm(uk) = 0 = wk by definition but this is a contradiction since wk cannot be 0 if it is linearly independent of the other wi's.

    From here, it seems like this same process can be applied backwards but I am not sure how it can be rigorously done in an elegant manner. I think I can use induction and say given that Tq(uk) is linearly independent from {Tr(uk): q + 1 ≤ r ≤ m, 1 ≤ k ≤ p} then Tq - 1(uk) is linearly independent for all k for if it wasn't then .... the same argument as the base case but applying repeated left multiples of T to keep creating 0 terms on the right eventually yielding the same contradiction of having wk = 0. This seems bulky and I am not confident it works. Also, I'm not use to using induction in a downwards trend and don't know if I would need to do an additional bottom base case for T1 specifically.

    Is this weird attempt at induction valid? Do you have any more elegant approaches? Thanks!

    For background, I am starting grad school in the fall and have to take a placement exam in linear algebra and vector calculus. This is a question off of one of the earlier placement exams.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 6, 2012 #2
    Re: Linear independent sets within repeated powers of a linear operator

    Your idea is right. But it is really... well.. backwards :)

    Say some Tiuk are linearly dependent. At least one of them has the minimal i. Does it give you any ideas?
     
  4. Aug 7, 2012 #3
    Re: Linear independent sets within repeated powers of a linear operator

    Thanks! I think that helps nicely. I was a little unsure that I would need to account for the linear combination involving Tq terms for q < i. But I think I see now that if Tqui is linearly independent then if it's coefficient wasn't 0 then we could rearrange it to show that it was indeed linearly dependent. Is that true? If you have a set of vectors and know that some are linearly independent of all the rest, then the linear dependence of any of the vectors can't involve a linearly independent vector? Thanks so much, I am sure I will keep posting questions the next few weeks.
     
  5. Aug 7, 2012 #4
    Re: Linear independent sets within repeated powers of a linear operator

    You have a linear combination, where some vectors will have a minimum i. Put them on the left side of the equation. The ones on the right will all correspond to higher degrees of the operator. Now apply the operator (m - i) times to the whole thing (that's your idea). What do you get?

    The only case when that can't work is when all i = m. But what do we have in that case?
     
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