(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose that T:W -> W is a linear transformation such that T^{m+1}= 0 but T^{m}≠ 0. Suppose that {w_{1}, ... , w_{p}} is basis for T^{m}(W) and T^{m}(u_{k}) = w_{k}, for 1 ≤ k ≤ p. Prove that {T^{i}(u_{k}) : 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 w_{1}, ..., w_{p}are linearly independent. Now suppose T^{m - 1}(u_{k}) 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 T^{m-1}(u_{k}) = c_{1}T^{m}u_{1}+ ... + c_{p}T^{m}u_{p}. If both sides are left multiplied by T then we have T^{m}= the linear combination of T^{m + 1}of the u_{i}'s which by definition are all 0. But then we have T^{m}(u_{k}) = 0 = w_{k}by definition but this is a contradiction since w_{k}cannot be 0 if it is linearly independent of the other w_{i}'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 T^{q}(u_{k}) is linearly independent from {T^{r}(u_{k}): q + 1 ≤ r ≤ m, 1 ≤ k ≤ p} then T^{q - 1}(u_{k}) 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 w_{k}= 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 T^{1}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

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# Homework Help: Linearly independent sets within repeated powers of a linear operator

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