R^n/Subspace of axes/Distance to point/Example

Let ${}J\subseteq \{1,\ldots ,n\}$, and let ${}U=\langle e_{i},\,i\in J\rangle \subseteq \mathbb {R} ^{n}$ denote the linear subspace spanned by this choice of standard vectors. Let

${}v={\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}}\,.$

Then, the distance of ${}v$ to ${}U$ equals

${}d(v,U)={\sqrt {\sum _{i\not \in J}v_{i}^{2}}}\,.$

The dropped perpendicular foot of ${}v$ on ${}U$ is

${}p_{U}(v)={\begin{pmatrix}w_{1}\\\vdots \\w_{n}\end{pmatrix}}\,,$

where

${}w_{i}={\begin{cases}v_{i},\,{\text{ if }}i\in J,\,\\0,\,{\text{ if }}i\notin J\,.\end{cases}}\,$