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