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Orthogonal projection is linear, and an orthonormal basis $(e_{i})$ of $W$ gives $P_{W} v = \sum_{i} ⟨ v, e_{i} ⟩ e_{i}$

Statement

Let $(e_{0}, \dots, e_{r - 1})$ be an orthonormal basis of a subspace $W$ of a finite-dimensional inner product space $V$ . Then

$P_{W} v = \sum_{i < r} ⟨ v, e_{i} ⟩ e_{i} .$

The map $P_{W}$ is linear, with image $W$ , kernel $W^{⊥}$ , and $P_{W}^{2} = P_{W}$ . Moreover, regarding both projections as endomorphisms of $V$ ,

$I - P_{W} = P_{W^{⊥}} .$

Facts & Assumptions

Given: A subspace $W$ , an orthonormal basis $(e_{i})_{i < r}$ of $W$ , and $v \in V$ .

[L1]

The orthogonal projection $P_{W} v$ is the unique $w \in W$ for which $v - w \in W^{⊥}$ (The orthogonal projection $P_{W} v$ is the $W$ -component in $V = W \oplus W^{⊥}$ ).

[L2]

In an orthonormal basis, the coefficient of a vector is its inner product with the corresponding basis vector (Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis).

[L3]

The inner product is linear in its first argument (Real and complex inner product spaces, with the inner product linear in the first argument).

Proof

technique · direct

1.1L2L3algebra

Put $p = \sum_{i < r} ⟨ v, e_{i} ⟩ e_{i}$ . Then $p \in W$ , and for every basis vector $e_{j}$ , [L3] and orthonormality give $⟨ v - p, e_{j} ⟩ = 0$ . By [L2], this makes $v - p$ orthogonal to all of $W$ .

1.2L1

The defining decomposition shows $P_{W} w = w$ for $w \in W$ and $P_{W} z = 0$ for $z \in W^{⊥}$ . Hence $im P_{W} = W$ , $\ker P_{W} = W^{⊥}$ , and $P_{W}^{2} = P_{W}$ .

2.1step 1.1L1L3

The uniqueness clause in [L1] gives $P_{W} v = p$ , proving the formula. Linearity follows immediately from [L3] and the formula.

3.1L1step 1.2∎

For the decomposition $v = P_{W} v + (v - P_{W} v)$ , the second summand lies in $W^{⊥}$ . Its projection onto $W^{⊥}$ is itself, so $P_{W^{⊥}} v = v - P_{W} v$ .

Depends on

Used by

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Sources

Sheldon Axler, Linear Algebra Done Right, 4th ed., result 6.57 (standard reference, not scraped)
Sergei Treil, Linear Algebra Done Wrong, Proposition 5.3.3 and Remark 5.3.4 (standard reference, not scraped)

Orthogonal projection is linear, and an orthonormal basis (ei)(ei​) of WW gives PWv=∑i⟨v,ei⟩eiPW​v=∑i​⟨v,ei​⟩ei​