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What's the relationship between the image and kernel of T and the image and kernel of T

^{n}? I think we saw in class something along the lines of:

[tex]Ker(T) \subseteq Ker(T^2)[/tex]

[tex]Im(T) \supseteq Im(T^2)[/tex]

My intuition is that this is also correct for any natural n, but is it true and if so how do you prove it, by induction?

Thanks,

Chen