I Don't really understand.. please help me. Suppose $A$ is a projection matrix on the plane $U$. For which plane is $I-A$ a projection matrix?
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$I-A$ is a projection on the null space of $A$, i.e. the space of vectors $x$ such that $Ax=0$. If this is in $3$ dimensions and $U$ is a $2$-dimensional plane, then $I-A$ is a projection on a line, not a plane.