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$f$ is a linear map on a $k$-vector space $M$ with $f^n=0$ for some $n>0$. $k$ is a field.

$k^d$ is $k$-algebra generated by $x_1,\ldots,x_d$ satisfy the relation $x_i x_j + x_j x_i =0$ for all $i,j=1, \ldots, d$.

Show that $\ker(f)$ is non-zero.

Thank you!

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    let x in k^d, then x = linear combination of x_i, then x-(sum r_ix_i) =0 then f (x-sum)= 0 ... but it does not work.. I do not know how can i use f^n =0 and the condition2011-02-27

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Hint: Choose $n$ in the question such that $f^{n-1}$ is not zero, and notice that $f(f^{n-1}(M))=\{0\}$.

Alternatively, prove that in general a composition of injective maps is injective.

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    @saba: If you like the answer (including the additional comment-it would appear so) please accept it by clicking the check mark next to this answer. It will prevent it from popping up in the future.2011-03-30