(a) Let x₁ = (0, -1,2), x₂ = = (1,0,−1), and x3 = (-3, 1,0). Apply the G-S procedure to produce three orthonormal vectors {91, 92, 93} in R³.
(a) Let x₁ = (0, -1,2), x₂ = = (1,0,−1), and x3 = (-3, 1,0). Apply the G-S procedure to produce three orthonormal vectors {91, 92, 93} in R³.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 46E
Related questions
Question
Graham Schmidt Orthogonalization
![[2] [Graham-Schmidt orthogonalization]
Recall the Graham-Schmidt (G-S) procedure from linear algebra; given
n linearly independent vectors {x₁, x2,...,n} and an inner product (...) on
that vector space, G-S produces a new set of mutually orthonormal vectors
{91, 92,...,n}, i.e.
i = j
i ‡ j.
The G-S process can be summarized by: For i = 1, 2, . . . n:
• Set v₁ = x1
. For i = 2,..., n set
. For i
=
2
(9i, 9j) =
i-1
Ui = Xi -Σ
j=1
.., n, normalize:
1,
0,
(f,g):
:=
qi =
where the norm ||v₁|| = ((vi, vi)) ¹/².
(a) Let x₁ = (0,−1, 2), x₂
=
(1,0,−1), and x3 = (-3,1,0). Apply the
G-S procedure to produce three orthonormal vectors {91, 92, 93} in R³.
(b) Consider the inner-product defined in lecture on functions:
(Xi, Vj)
-Vj
(Vj, vj)
[
[f
Vi
||v₂||
f(x)g(x)dx.
Consider the 'vectors' in this function space {y₁ (x), Y2(x), Y3(x)} = {1, x, x²}.
Perform the G-S procedure to produce three orthonormal polynomials {q₁ (x), 92(x), 93 (x)};
this will produce the first three Legendre polynomials.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F96308306-b7f4-4349-b305-1094d37e82f2%2F0f4ck4_processed.png&w=3840&q=75)
Transcribed Image Text:[2] [Graham-Schmidt orthogonalization]
Recall the Graham-Schmidt (G-S) procedure from linear algebra; given
n linearly independent vectors {x₁, x2,...,n} and an inner product (...) on
that vector space, G-S produces a new set of mutually orthonormal vectors
{91, 92,...,n}, i.e.
i = j
i ‡ j.
The G-S process can be summarized by: For i = 1, 2, . . . n:
• Set v₁ = x1
. For i = 2,..., n set
. For i
=
2
(9i, 9j) =
i-1
Ui = Xi -Σ
j=1
.., n, normalize:
1,
0,
(f,g):
:=
qi =
where the norm ||v₁|| = ((vi, vi)) ¹/².
(a) Let x₁ = (0,−1, 2), x₂
=
(1,0,−1), and x3 = (-3,1,0). Apply the
G-S procedure to produce three orthonormal vectors {91, 92, 93} in R³.
(b) Consider the inner-product defined in lecture on functions:
(Xi, Vj)
-Vj
(Vj, vj)
[
[f
Vi
||v₂||
f(x)g(x)dx.
Consider the 'vectors' in this function space {y₁ (x), Y2(x), Y3(x)} = {1, x, x²}.
Perform the G-S procedure to produce three orthonormal polynomials {q₁ (x), 92(x), 93 (x)};
this will produce the first three Legendre polynomials.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning