can be expressed uniquely in the form 
where 
is a scalar multiple of 
and 
is a scalar multiple of 
Find the matrix 
such that![\[\mathbf{P} \mathbf{v} = \mathbf{a}\]](https://latex.artofproblemsolving.com/1/1/f/11f8c77feb79afecfff831f7ec8e78412698a812.png)
for all vectors 
Since P ( 3 1)T should be zero, a good starting point is to find the projection operator, PP that will project ( 3 1)T out of any vector v.
This operator is PP = 1 - outer product of ( 3 1)T with itself divided by the inner product of
( 3 1 )T with itself. This works out to: PP =
1/10 -3/10
-3/10 9/10
It is easy to verify that PP ( 3 1)T is the null vector.
The matrix, P , that we want is proportional to ( 2 -1) PP ,
That is, proportional to the row vector ( 1/2 -3/2 )
Since we want P (2 -1)T = 1 , the proportionality factor is 2/5
The final form of P is thus: ( 1/5 -3/5)
