Consider a linear operator τ:V→V (where V is finite-dimensional) such that rk τ(adsbygoogle = window.adsbygoogle || []).push({}); ^{2}=rk τ. Show that (im τ) [itex]\cap[/itex] (ker τ) is the zero space.

Here's where I am:

Its easy to see that im τ=im τ^{2}, since it is a subspace with the same dimension. I also know that if (im τ) [itex]\cap[/itex] (ker τ) contains a nonzero vector, then it has positive dimension.

My intuition is that I can use that positive dimension and the rank plus nullity theorem to show a contradiction, but I just can't seem to figure out how. Rank-plus-nullity gives me null τ = null τ^{2}, and I know that dim ((im τ) [itex]\cap[/itex] (ker τ)) ≤ null τ. Any idea where to go next?

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# Operator powers

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