Let P_{W} denote the projection matrix for the subspace spanned by W, then

(1)  Ŷ = P_{W} Y = W α ∈^{(} W).

In other words, the estimates of Y are a linear combination of columns of W. Then we can write

(2)  Ỹ = (1 − P_{W})Y ∈^{(} W^{⊥}).

Ỹ is the vector of residuals from regressing W onto Y. Y can then be expressed as

(3)  Y = Ŷ + Ỹ