Projection Example 1

>> X = [[1;2;3],[1;0;1]]
X =

   1   1
   2   0
   3   1

>> PX = X * (X' * X)^(-1) * X'
PX =

   0.66667  -0.33333   0.33333
  -0.33333   0.66667   0.33333
   0.33333   0.33333   0.66667

>> y = [2;4;6]
y =

   2
   4
   6

>> yX = PX * y
yX =

   2.0000
   4.0000
   6.0000

Since y in this example already lies in the column-space of X, orthogonal projection onto that space has no effect.