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

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product, and let ${}U\subseteq V$ denote a linear subspace with an orthonormal basis ${}u_{1},\ldots ,u_{m}$ of ${}U$.

Then the

orthogonal projection onto ${}U$ is given by

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

Proof, Write another proof