Page 246 - A Course in Linear Algebra with Applications

230 Chapter Seven: Orthogonality in Vector Spaces

Apply 7.3.3 with si = X\ and s 2 = X 2\ we find that the
projection of X on S is

P = <X,Xi >Xi+<X,X 2 >X 2

4 1 1 / 16>

Having seen that orthonormal bases are potentially useful, let
us now address the problem of finding such bases.

Gram-Schmidt orthogonalization
Suppose that V is a finite-dimensional real inner prod-
uct space with a given basis {ui,... u n }; we shall describe
,
a method of constructing an orthonormal basis of V which is
known as the Gram-Schmidt process.
The orthonormal basis of V is constructed one element
at a time. The first step is to get a unit vector;

1
Vi = j . jj-Ui.

Notice that ui and vi generate the same subspace; let us call
it Si. Then vi clearly forms an orthonormal basis of Si. Next
let
Pi = < u 2 , vi > vi.
By 7.3.3 this is the projection of 112 on Si. Thus u 2 — p x
belongs to S^ and u 2 — Pi is orthogonal to vi. Notice that
U2 ~ Pi ^ 0 since ui and u 2 are linearly independent. The
second vector in the orthonormal basis is taken to be

V2 = 71 n-(u 2 - P i ) .
U
ll 2-Pll|
By definition of vi and v 2 , these vectors generate the same
subspace as Ui, 112, say S 2. Also vi and v 2 form an orthonor-
mal basis of £2 •