Linear Algebra: UNIT III: Inner Product Spaces
The Gram Schmidt Orthogonalization: Theorems Part 4
Important Theorems for Engineering Maths or Mathematics - The Gram Schmidt Orthogonalization: Theorems Part 4
THE GRAM
SCHMIDT ORTHOGONALIZATION
THEOREMS PART 4
Theorem 9
Projection theorem:
Let W be a finite
dimensional subspace of an inner product space V and let y ∈ V. Then there exists unique
vectors u ∈
W and z ∈
W┴ such that y=u+z. Furthermore, if {v1, v2, ..., vk
} is an orthonormal basis for W, then
Proof:
Let
{ v1, v2, ... vk } be an orthonormal basis for
W.
Let
u be as defined by
and let z=y−u.
Clearly
u ∈ W and y=u+z
To
show that z ∈
W┴.
(i.e.) z is orthogonal to (every element in W)
each vj For any j we have
< z vj> = < y − u, vj
>
=
<y,vj> − <y,vj> ||vj||2
=
<y, vj> − <y,vj> (1)
=
0
< z, vj > = 0
⇒ z ∈ W┴.
To
show uniqueness of u and z
Suppose
that y=u+z = u'+z' where u' ∈
W, z' ∈ W┴.
Then
u−u' = z'−z ∈
W∩W┴
u
– u' = z' − z ∈
W∩W┴ = { 0 }
u= u' and z = z'
u, u' ∈
W and z, z' ∈
W┴
u−u' ∈
W, z−z' ∈ W┴.
Theorem 10
The vector u is the
unique vector in W, that is "closest" to y. (i.e) for any x ∈ W;
|| y − x || ≥ || y − u || and this inequality
is an equality if x=u.
Proof:
From
the previous theorem we have y = u+z, where u ∈ W, and z ∈ W┴.
Let
x ∈ W then u−x ∈ W. [ W is a subspace]
u−x is orthogonal to z.
[ z is perpendicular to every element in W ]
||y − x||2 = || u + z − x ||2
=
|| (u − x) + z ||2
=
|| x − x ||2 + || z ||2
≥ || z ||2 = || y − u ||2
||
y − x || ≥ || y − u ||
Suppose
that, ||y−x| = ||y−u||
|| y − x ||2 = || y − u ||2
|| u + z − x ||2 = || u + z − u ||2
||
u + z − x ||2 = || z ||2
||u−x||2 + ||z||2 = ||z||2
||u−x||2 = 0
||u−x|| = 0
⇒ u−x = 0
u = x.
Note:
The vector u in the above theorem is called the orthogonal projection of y on w1.
Linear Algebra: UNIT III: Inner Product Spaces : Tag: maths, mathematics : - The Gram Schmidt Orthogonalization: Theorems Part 4
Linear Algebra: UNIT III: Inner Product Spaces
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