Vector space/K/Inner product/Finite dimension/Linear subspace/Orthogonal projection/Orthonormal basis/Fact/Proof

Proof

We extend the basis to an orthonormal basis ${}u_{1},\ldots ,u_{n}$ of ${}V$. The orthogonal complement of ${}U$ is

${}U^{\perp }=\langle u_{m+1},\ldots ,u_{n}\rangle \,.$

Due to fact, we have

${}v=\sum _{i=1}^{n}\left\langle v,u_{i}\right\rangle u_{i}={\left(\sum _{i=1}^{m}\left\langle v,u_{i}\right\rangle u_{i}\right)}+{\left(\sum _{i=m+1}^{n}\left\langle v,u_{i}\right\rangle u_{i}\right)}\,.$

Therefore, ${}\sum _{i=1}^{m}\left\langle v,u_{i}\right\rangle u_{i}$ is the projection onto ${}U$ along ${}U^{\perp }$.

To fact