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TranscendArcu
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Homework Statement
http://desmond.imageshack.us/Himg810/scaled.php?server=810&filename=screenshot20120131at923.png&res=medium
The Attempt at a Solution
So in particular I want to look at the last part of this problem. That is, "Show that [itex]S^n = 0[/itex]"
I know that [itex]dim(KerS^k) = k[/itex] and therefore, [itex]dim(ImS^k)= n-k[/itex]. If k=n, [itex]dim(ImS^k)= n- n = 0[/itex] which implies that [itex]ImS^k = \left\{ 0 \right\} [/itex].
I'm having trouble drawing from this the conclusion that [itex]S^n = 0[/itex]. Is S an isomorphism? If so, does that mean that the only thing that can map to 0 in the codomain is 0 in the domain?
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