= Oblique symmetry of complementary subspaces
{title2=$T(u+w)=u-w,\quad V=U\oplus W$}
For complementary <vector subspaces> $U,W$ of a real <vector space>, the rule $T(u+w)=u-w$ is a well-defined <linear map> satisfying $T^2=I$. It fixes $U$ pointwise and reverses $W$. In a <basis> formed by adjoining <bases> of $U$ and $W$, its <matrix> is $\operatorname{diag}(I,-I)$. Conversely every real <linear map> with $T^2=I$ has this form, with $U=\ker(T-I)$ and $W=\ker(T+I)$, as follows from $v=(v+Tv)/2+(v-Tv)/2$.
In a <Euclidean space>, such a <linear map> is an <orthogonal transformation> if and only if $U$ and $W$ are perpendicular. Necessity follows from $\langle u,w\rangle=\langle Tu,Tw\rangle=-\langle u,w\rangle$; sufficiency follows by expanding the squared <Euclidean norm> of $u+w$. In three dimensions, a fixed plane with reversed complementary line is a possibly oblique reflection, and a fixed line with reversed complementary plane is a possibly oblique axial symmetry. Their diagonal forms alone do not imply a Euclidean <reflection in a hyperplane> or a Euclidean half-turn.
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