Discrete Mathematical Structures by Pearson

12-56 Discrete Mathematical Structures

Forming inner product with



x

k

,

we get

cxxcxx cxxcxx

kkkkknkn1122

, ,, , 0

<>+< >+ +< >+ +< >=

 







 

Since the given vectors are orthogonal, each of these inner products is zero except

<>.



xx

kk

,

Hence, we have

cx

xc

kk

kk

<>=⇒ =



,0 0

Giving values k = 1, 2,…, n in turn, we obtain c

1

= c

2

= … = c

n

= 0.

Therefore, the given set of vectors is linearly independent.

12.8.6 Gram–Schmidt Orthogonalization Process

Let V be a vector space whose basis is given, that is, a set of linearly independent vectors is given.

From the given set, we can construct an orthogonal set. The process is known as Gram–Schmidt

orthogonalization, ...