information:extraction, information:filtering, information:classification
Projection

An important example is the projection parallel to some direction onto an affine subspace. The importance of this example lies in the fact that Euclidean spaces are affine spaces, and that these kinds of projections are fundamental in Euclidean geometry.

More precisely, given an affine space E with associated vector space E → {\displaystyle {\overrightarrow {E}}}, let F be an affine subspace of direction F → {\displaystyle {\overrightarrow {F}}}, and D be a complementary subspace of F → {\displaystyle {\overrightarrow {F}}} in E → {\displaystyle {\overrightarrow {E}}} (this means that every vector of E → {\displaystyle {\overrightarrow {E}}} may be decomposed in a unique way as the sum of an element of F → {\displaystyle {\overrightarrow {F}}} and an element of D). For every point x of E, its projection to F parallel to D is the unique point p (x) in F such that

This is an affine homomorphism whose associated linear map p → {\displaystyle {\overrightarrow {p}}} is defined by

for x and y in E.

The image of this projection is F, and its fibers are the subspaces of direction D.