Prove that there is one and only one vector satisfying the equations
where and is orthogonal to in .
Proof. First, we show uniqueness. If is any vector such that and and is another vector such that and then we have
But, by a previous exercise (Section 13.11, Exercise #8) we know these conditions imply . Hence, there is at most one such .
Now, for existence. Let . Since are orthogonal we know ,
From we then have
Since , we know at least one of the . Without loss of generality, assume . From the second and third equations we have
Hence, can take any value. The vectors such that are then of the form
Since (since at least one of the , we have that such a vector always exists