Linear Algebra: UNIT III: Inner Product Spaces
The Gram Schmidt Orthogonalization: Theorem Part 3
Important Theorems for Engineering Maths or Mathematics - The Gram Schmidt Orthogonalization: Theorem Part 3
THE GRAM
SCHMIDT ORTHOGONALIZATION
THEOREM PART 3
Theorem 8
Let
V = P2(R) with the inner product <f(x), g(x)> = 0ʃ1 ƒ(x)g(x) dx and consider
the subspace P2(R) with the standard ordered basis B use the Gram−Schmidt
Process to replace B by an orthogonal basis {v1, v2, v3}
for P2(R) and then use this orthogonal basis to obtain an
orthonormal basis for P2(R) and write h(x)=1+x as a linear
combination of orthonormal basis.
Definition
Orthogonal complement:
Let
S be a non−empty subset of an inner product space V. We define S┴("S
perp") to be the set of all vectors in V that are orthogonal to every
vector in S.
(i.e.)
S┴ = { x ∈
V; such that <x, y> = 0 for all y ∈
S}
The
set S┴ is called the orthogonal complement of S.
Note
that S┴ is a subspace of V for any subset S of V.
The
following three examples explain the above concept of orthogonal complement.
(i)
Let {0}┴ = V
clearly
{0}┴ 
V
Let
x ∈ V
⇒ <x, 0> = 0
x = { 0 }┴
V {0}┴
V = { 0
}┴
(ii)
V┴ = {0}
Let
x ∈ V┴
⇒ <x, y>=0 for all
y ∈ V.
In
particular y=x.
<x, x> = 0
⇒
x=0
V = { 0 }
(iii)
W∩W┴ = {0}.
Let
x ∈ W∩W┴
(i.e.)
x ∈ W and x ∈ W┴
(x, y) = 0 for all y ∈ W.
In
particular y=x
<x, x> = 0
⇒ x=0
W∩W┴ = {0).
Linear Algebra: UNIT III: Inner Product Spaces : Tag: maths, mathematics : - The Gram Schmidt Orthogonalization: Theorem Part 3
Linear Algebra: UNIT III: Inner Product Spaces
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