Science:Math Exam Resources/Courses/MATH307/April 2005/Question 03 (b)/Solution 1

Why is $PQ=0$

$Px$ is in the subspace$V$ and $Qx$ will be in the subspace orthogonal to $V$. So it is easy to imagine that if $y$ is in $V$ then $Qy$ will be the zero vector meaning $Qy=0$. To get a vector in $V$ from any vector in$R^n$ we simply use $y=Px$. Substituting this into the first equation we get $Q(Px)=QPx=0$ and because x does not have to be a zero vector $QP=0$. The same logic is true for the opposite case so $PQ=0$.

To see RQ = Q we can solve algebraically

$RQ=(I-2P)Q=Q-2PQ$

but as seen above $PQ=0$

so $Q-2PQ=Q$