A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.

For finite-dimensional $W$ in an inner-product space, every $v$ has a unique decomposition $v=w+z$ with $w\in W$ and $z\in W^\perp$. Set $w=\sum_iq_i\langle q_i,v\rangle$ using an orthonormal basis of $W$. Uniqueness follows from $W\cap W^\perp=\{0\}$.Copy