The orthogonal projection (or simply “projection”) of onto is defined by
The complex scalar is called thecoefficient of projection. When projecting onto a unit length vector , the coefficient of projection is simply the inner product of with .
Motivation: The basic idea of orthogonal projection of onto is to “drop a perpendicular” from onto to define a new vector along which we call the “projection” of onto . This is illustrated for in Fig. 6.9 for and , in which case
Derivation: (1) Since any projection onto must lie along the line colinear with , write the projection as . (2) Since by definition the projection is orthogonal to , we must have