Consider a linear operator τ:V→V (where V is finite-dimensional) such that rk τ2=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 τ2, 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 τ2, and I know that dim ((im τ) [itex]\cap[/itex] (ker τ)) ≤ null τ. Any idea where to go next?