The orthogonal projection gives
(1)
The first summand has every column in the column space of , and the second has every column in its orthogonal complement. They therefore have zero Frobenius inner product. The Pythagorean identity, together with , gives the stronger exact formula
(2)
Consequently
(3)
with equality exactly when . Thus solving the linear least-squares problem for each column chooses the best coefficients in the prescribed column space.

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