## Images of Powers of Transformations

For some technical points, it’s going to be useful to have a sort of dual to the increasing chain of subspaces we found yesterday. Instead of kernels, we’ll deal with images.

Specifically, if then I say . Indeed, the first statement asserts that there is some so that . But then , and so it’s the image of under as well. So we have a decreasing sequence

Just like last time, these stabilize by the time we get to the th power, where . Instead of repeating everything, let’s just use the rank-nullity theorem, which says for each power that . Now if then we calculate

where in the second line we used the stability of the sequence of kernels from yesterday. This tells us that for all these higher powers of .

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[…] the dimension of is , and so by this point the sequence of images of powers of has stabilized! That […]

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[…] the last equality holds because the dimension of is , and so the image has stabilized by this point. Thus we can choose so that . And […]

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