= Symmetrization map in characteristic two
{title2=$T(A)=A+A^T,\quad\operatorname{char}F=2$}
Over a <field> of <characteristic two>, the <linear map> $T:M_n(F)\to M_n(F)$, $A\mapsto A+A^T$, has <kernel of a linear map> the <symmetric matrices>, of <dimension> $n(n+1)/2$. Its <image of a linear map> is the <linear subspace> of <symmetric matrices> with zero diagonal, of <dimension> $n(n-1)/2$. Indeed, $T(A)=0$ means $A=A^T$, and every symmetric zero-diagonal <matrix> is the image of the <matrix> obtained by retaining just its entries above the diagonal. Thus the two <dimensions> still add to $n^2$, as required by the <rank-nullity theorem>.
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