### Projection

The *orthogonal projection* (or simply ``projection'') of
onto
is defined by

*coefficient 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.5.9 for and
, in which case

**Derivation:** (1) Since any projection onto must lie along the
line collinear with , write the projection as
. (2) Since by definition the *projection error*
is orthogonal to , we must have

Thus,

See §I.3.3 for illustration of orthogonal projection in matlab.

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The Pythagorean Theorem in N-Space