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Orthogonal projection onto a finite-dimensional subspace

If

e_{0}, \dots, e_{n}

is an orthogonal basis of a finite-dimensional subspace

W

, the unique closest point in

W

to

f

is

P_{W} f = \sum_{k = 0}^{n} \frac{⟨ f , e _{k} ⟩}{⟨ e _{k} , e _{k} ⟩} e_{k} .

The residual

f - P_{W} f

is orthogonal to

W

.

Table of contents 88 2
- Orthogonal projection matrix Orthogonal projection onto a finite-dimensional subspace 18
- Least-squares polynomial in an orthogonal-polynomial basis Orthogonal projection onto a finite-dimensional subspace 38

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