Principles of Mathematics in Operations Research

40 3 Orthogonality

Pa=(a,b,0)

Fig. 3.4. Orthogonal projection

Remark 3.2.9 When we find an orthogonal basis that spans the ground vec-

tor space and the coordinates of any vector with respect to this basis is on

hand, the projection of this vector into a subspace spanned by any subset of

the basis has coordinates 0 in the orthogonal complement and the same coordi-

nates in the projected subspace. That is, the projection operation simply zeroes

the positions other than the projected subspace like in the above example. One

main aim of using orthogonal bases like E = {ej}™=1 for the Cartesian sys-

tem, W^1 , is to have the advantage of simplifying projections, besides many

other advantages like preserving lengths.

Proposition 3.2.10 Multiplication by an orthogonal Q preserves lengths

\\Qx\\ = \\x\\, \fx;

and inner products

(Qx)T(Qy)=xTy,Vx,y.

3.2.2 Gram-Schmidt Orthogonalization

Let us take two independent vectors a and b. We want to produce two per-

pendicular vectors v\ and v 2 :

, >

V

T° r

vi = a, v 2 = b — p = b Tp — vi =>• v{ v 2 = 0 =>• vi ± v 2.

v(vx

If we have a third independent vector c, then

V3 = C 7f Vivi c fvA V c 2 => V 3 -L V 2 , V 3 ±Vi.

V{ V\ V$ V 2

If we scale Vi,v 2 ,v 3 , we will have orthonormal vectors:

Vi

Qi = ii—M, a 2

v 2 v 3

«2^93 = "3