If w is a subspace of r" and if is in both w and w, then must be the zero vector in the orthogonal decomposition theorem, each term in y is itself an orthogonal projection of f onto a subspace of w. lf у z1 +z2, where ź1 is in a subspace w and z2 is in w , then z1 must be the orthogonal projection of y onto w the best approximation to y by elements of a subspace w is given by the vector y-projwy. if an n p matrix u has orthonormal columns, then uutx= for all x in rn