Principles of Mathematics in Operations Research

3.2 Projections and Least Squares Approximations 41

Proposition 3.2.11 Any set of independent vectors ai,a,2,- • • ,an can be con-

verted into a set of orthogonal vectors v\, V2, • • •, vn by the Gram-Schmidt pro-

cess. First, Vi = a\, then each Vi is orthogonal to the preceding v\, v-i,..., «i_i:

Vi = a,-

vj a%

vfvi

Vi

ui-lui

Vi-l.

For every choice of i, the subspace spanned by original ai,a2,-..,aj is also

spanned by v\, v-i,..., Vi. The final vectors

{*

=

i&}

Vj_

«illJi=i

are orthonormal.

Example 3.2.12

vi = a%, and

a~ v\ 1

v(vi 2 *

Let

a 2 -

ai =

\vi --

"1"

0

1

, a 2 =

1

1

0

, as =

0

1

1

= f i => v 3 = a 3 iui \v 2 = Then,

q\

and 03

l

V2

"1"

0

1

r! i

V2

(^0) l ?2

• 1 •
  2
  1
  1
  2.

1 _

2

vG

1

. V6.

9

12

• 2 "
  3
  2
  3
  2
  . 3.

—

1 "

V3

1

v/3

1

. vs.

Ol i>i = -v/2oj

«2 = §«i + «2 = \J\qi +

as = \vx + \v 2 + v 3 = yj\qi + yj\q2 + yj §<

2°2

<£> [ai,a 2 ,a 3 ] = [gi, 02,03]

»3

<^> A = QR.

Proposition 3.2.13 A — QR where the columns of Q are orthonormal vec-
tors, and R is upper-triangular with \vi\ on the diagonal, therefore is invert-
ible. If A is square, then so are Q and R.