Principles of Mathematics in Operations Research

36 3 Orthogonality

Remark 3.1.13 The following statements are equivalent,

i. W^V^1.

a v = w

±

Hi. W ± V and dimV + dimW — n.

Proposition 3.1.14 The following are true:

i. N{AB)2M{B).

ii. Tl(AB) C 11(A).

iii.Af((AB)T)DAf(AT).

iv. Tl{(AB)T) C Tl{BT).

Proof. Consider the following:

i. Bx = 0 => ABx = 0. Thus, Vx € M(B), x £ Af(AB).

ii. Let b 3 ABx = b for some x, hence 3y = Bx 3 Ay = b.

iii. Items (iii) and (iv) are similar, since (AB)T = BTAT. O

Corollary 3.1.15

rank(AB) < rank(A),

rank(AB) < rank(B).

3.1.3 Angle between two vectors

See Figure 3.2 and below to prove the following proposition.

c = b — a =$• cos c = cos(6 — a) = cos b cos a + sin b sin a

— JiL J^L 4. Jf2__^2_ _ "1^1 + U2V2

cosc

~ U\U\ U\ Nl " IHIIHI "

I

X-Axis

U=(U Lfe)

v=(v„v 2 )

Fig. 3.2. Angle between vectors