Soft Sensing - Advanced Control and Supervision of Mineral Processing Plants - page 181

y , x = y , x x u = x y , x u = x y , x u − x u , x u . (4.85)

By replacing (4.81) in (4.85), and by (4.82)

x y , x u x u , x u = x y , x u = y , x x u = y , x . (4.86)

Hence, by (4.84), (4.85) and (4.86) y − y , x = 0.

y

x

x , y

y x

=

2

x

Figure 4.10 Projection y of y onto x , which may be vectors in Euclidean space, or elements in other
inner product spaces such as those in Table 4.6

A general Pythagoras theorem holds in inner product spaces. Let x ⊥ y , i.e. ,

y , x = 0, and let z = x + y .Then

2 2 2

z = z , z = x + y , x + y = x , x + x , y + y , x + y , y = x + y (4.87)

Then the Pythagoras theorem in Euclidean spaces is just a particular case of the
Pythagoras theorem in inner product spaces (and in Hilbert spaces in general).

4.3.1.3 Projection on a Subspace

= { , ,..., } ⊂

Let B m ϕ 1 ϕ 2 ϕ m be a base for a subspace V m ℵof the inner product

∈

space ℵ. Therefore, any element h ℵ may be written as the linear combination

m
k = 1 α k ϕ k .

h = (4.88)

Let

= { , ,..., }

B mon ζ 1 ζ 2 ζ m (4.89)