5.2 Orthogonal Complements and Projections

$ 5 .2 # 1

5.2 Orthogonal Complements and projections Orthogonal complements Definition Let W  be a subspace of  R n and let x

 n

R

.

(a) x is orthogonal to W  Ê  x  W  Ê w W  x  w  Ê w W  x  w  0 . (b) W   the orthogonal complement of  W   n  the set of all vectors in R that are orthogonal to W .  n  n   x R : x  W    x R : w W  x  w    n   x R : w W  x  w  0  .

Theorem (5.9) Let W  be a subspace of  R n . Then: (a) W  is also a subspace of  R n . (b) W W ,   0 . (c) if  W   span w 1 , w 2 , , w k  then  n W   v R :  v  w i  0  for i  1, 2,

, k .

1. Proof (a) . W , is not empty: W  0

w



w



0

Í

0

W 

Í

W 

. W , is closed closed under addition addition:

W  and y

 x

W 

Í

w

W  x



w



Í

w

W   x



y w

Í

x



y

0  and w 

 x

W  y 



w    y

W  .

. W , is closed under scalar multiplication:

0

w





w



00



0