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Is it true or false that

if $T:V\to V$ is linear transformation such that $T^3+I=0$, then $\dim V\geq 3$?

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No, if $I$ is the identity, then $-I:V\to V$ satisfies the relation $(-I)^3+I=0$, with no assumption on the dimension of $V$. –  detnvvp Jul 26 '13 at 11:28
    
The statement is false. –  Tunococ Jul 26 '13 at 11:28
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1 Answer 1

up vote 4 down vote accepted

No, if $T=-I$ in any vector space, in particular in ones of dimension${}<3$, then $T^3+I=0$ will hold.

Besides, you can never get a lower bound on the dimension from an identity that you linear map satisfies, because in dimension$~0$ any identity you could write down will hold (as there is only one linear map).

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