Page 247 - A Course in Linear Algebra with Applications

7.3: Orthonormal Sets and the Gram-Schmidt Process 231

The next step is to define

p 2 = < u 3 , vi > vi+ < u 3 , v 2 > v 2 ,

which by 7.3.3 is the projection of u 3 on S^. Then u 3 — p 2
belongs to S^ and so it is orthogonal to vi and v 2 . Again
one must observe that u 3 — p 2 7^ 0, the reason being that Ui,
u 2 , u 3 are linearly independent. Now define the third vector
of the orthonormal basis to be

V 3 = U3
Ti H ( ~ Ps)-
I
IIU3-P2II
Then vi, v 2 , v 3 form an orthonormal basis of the subspace
generated by ui, u 2 , u 3 .
5 3
The procedure is repeated n times until we have con-
structed n vectors v i , . . . , v n ; these will form an orthonormal
basis of V.
Our conclusions are summarised in the following funda-
mental theorem.

Theorem 7.3.4 (The Gram - Schmidt Process)
Let {ui,..., u n } be a basis of a finite-dimensional real inner
product space V. Define recursively vectors v i , . . . , v n by the
rules

u
vi = vi—[7U1 and v i + i = ir( i+i _ Pi)'
u
u
ll l|| ll i+l-Pill
where

Pi = < Ui+i. v i > v i H h < u i + i , Vi > Vj
is the projection of u i+i on the subspace Si =< v i , . . . , Vj >.
Then v i , . . . , v n form an orthonormal basis ofV.

The Gram-Schmidt process furnishes a practical method
for constructing orthonormal bases, although the calculations
can become tedious if done by hand.