Introduction to Computational Linear Algebra

32 Introduction to Computational Linear Algebra

Consequently, by induction, one proves that such process leads to a new or-

thonormal set {f

i

|i = 1, ..., n}, i.e.,

(f

i

)

T

(f

j

) = 0, i 6= j and (f

i

)

T

(f

i

) = 1, ∀i,

in which one obtains [f

1

, f

2

, ..., f

m

] from [v

1

, v

2

, ..., v

m

] using the relations:

v

1

= c

1

f

1

v

2

= c

2

f

2

+ (v

T

2

f

1

)f

1

..........

v

i

= c

i

f

i

+ (v

T

i

f

1

)f

1

+ ... + (v

T

i

f

i−1

)f

i−1

......

v

m

= c

m

f

n

+ (v

T

m

f

1

)f

1

+ ... + (v

T

m

f

n−1

)f

m−1

.

(2.2)

Since c

i

6= 0, i = 1, ...m then span(F ) = span(V ) = V

Figure 2.1 shows the first steps of a Gram-Schmidt process transforming a

linearly independent set {U

1

, U

2

} into an orthonormal set {v

1

, v

2

}.

y

x

0

proj v

1

u

2

u

1

u

2

v

2

v

1

FIGURE 2.1: A Gram-Schmidt ...