Orthogonal projection

Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $W$ be a finite-dimensional subspace of $V$. By the orthogonal decomposition, each $v \in V$ is $v = w + w'$ with unique $w \in W$ and $w' \in W^\perp$. The orthogonal projection onto $W$ is the map $P_W \colon V \to V$, $P_W(v) = w$.

Remarks

$P_W$ is a linear projection with image $W$ and kernel $W^\perp$, and it is self-adjoint. For an orthonormal basis $e_1, \dots, e_k$ of $W$, $P_W(v) = \sum_{i=1}^{k} \langle v, e_i \rangle e_i$. For the line spanned by $u \ne 0$, $P(v) = \frac{\langle v, u \rangle}{\langle u, u \rangle} u$. $P_W(v)$ is the point of $W$ closest to $v$.

Explanations

L1 human

The orthogonal projection drops a perpendicular from a vector onto a subspace. The foot of the perpendicular is the projection. In $\R^3$, projecting onto the $xy$-plane sets the $z$-coordinate to zero.