Projection

Definition

Let $F$ be a field and let $V$ be a vector space over $F$. A projection on $V$ is an endomorphism $P \colon V \to V$ with $P \circ P = P$.

Remarks

If $P$ is a projection, then $V = \im P \oplus \ker P$, and $P$ acts as the identity on $\im P$. Conversely, for each direct sum $V = U \oplus W$, the map $u + w \mapsto u$ is a projection with image $U$ and kernel $W$. If $P$ is a projection, then $I - P$ is the projection with image $\ker P$ and kernel $\im P$. In an inner product space, an orthogonal projection is a projection whose kernel is orthogonal to its image.

Explanations

L1 human

A projection maps space onto a subspace and leaves the vectors of that subspace where they are. Projecting a second time changes nothing. The shadow of an object on the floor, in light from a fixed direction, is a picture of a projection.