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3. Numerical Linear Algebra 101

T
T
T
T
q= v3- [(o1 v3) / (o1 o1)] o1] - [(o2 v3) / (o2 o2)] o2]

o3 =q / ||q||
5.2 Example

Consider the set of independent column vectors v1, v2 and v3 as displayed
below.

1 2 4
v1= 2 v2= 1 v3 = 2
3 4 3

The corresponding set of orthonormal vectors computed using Gram-
Schmidt orthogonalization procedure is given below.

0.2673 0.5203 0.8111
o1 = 0.5345 o2 = -0.7804 o3 = 0.3244

0.8018 0.3468 -0.4867
The dot product of the vectors o1, o2 and o3 is displayed in the matrix
form for illustration.

1.0000 -0.0000 0.0000
-0.0000 1.0000 0.0000
0.0000 0.0000 1.0000

Note that the matrix obtained is the identity matrix as expected.

5.3 Need for Orthonormal Basis

Suppose the vector ‘V’ in the vector space ‘S’ is represented as the linear
combination of the orthonormal basis computed in the section 5.1

9
V = 6
14

The co-efficients are obtained as the inner product of the vector ‘V’ and
the corresponding orthonormal basis.