*Problem 3.4 Suppose you start out with a basis (lei), le2), ..., len)) that is not orthonormal. The Gram-Schmidt procedure is a systematic ritual for generating from it an orthonormal basis (le), le2), ..., le)). It goes like this: (i) Normalize the first basis vector (divide by its norm): le) = le₁) lleill (ii) Find the projection of the second vector along the first, and subtract it off: le₂) - (ele₂) lei). This vector is orthogonal to le); normalize it to get le₂). (iii) Subtract from le3) its projections along le₁) and le₂): les) - (ejles) lej) - (e₂lez)|e₂). This is orthogonal to le) and le₂); normalize it to get leg). And so on. Use the Gram-Schmidt procedure to orthonormalize the three-space basis |e₁) = (1 + i)i + (1)ĵ+ (i)k, \e₂) = (i)î+ (3)ĵ+(1)k, |e3) = (0)î+ (28)ĵ+ (0)k.
*Problem 3.4 Suppose you start out with a basis (lei), le2), ..., len)) that is not orthonormal. The Gram-Schmidt procedure is a systematic ritual for generating from it an orthonormal basis (le), le2), ..., le)). It goes like this: (i) Normalize the first basis vector (divide by its norm): le) = le₁) lleill (ii) Find the projection of the second vector along the first, and subtract it off: le₂) - (ele₂) lei). This vector is orthogonal to le); normalize it to get le₂). (iii) Subtract from le3) its projections along le₁) and le₂): les) - (ejles) lej) - (e₂lez)|e₂). This is orthogonal to le) and le₂); normalize it to get leg). And so on. Use the Gram-Schmidt procedure to orthonormalize the three-space basis |e₁) = (1 + i)i + (1)ĵ+ (i)k, \e₂) = (i)î+ (3)ĵ+(1)k, |e3) = (0)î+ (28)ĵ+ (0)k.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:*Problem 3.4 Suppose you start out with a basis (le1), le₂),..., len)) that is not
orthonormal. The Gram-Schmidt procedure is a systematic ritual for generating
from it an orthonormal basis (le), le₂),..., len)). It goes like this:
(i) Normalize the first basis vector (divide by its norm):
le₁) =
le₁)
lleill
(ii) Find the projection of the second vector along the first, and subtract it off:
le₂) - (eile₂) lei).
This vector is orthogonal to le); normalize it to get le₂).
(iii) Subtract from e3) its projections along le) and le₂):
les) - (ejles) lej) - (e₂|e3)|e2).
This is orthogonal to le) and le2); normalize it to get leg). And so on.
Use the Gram-Schmidt procedure to orthonormalize the three-space basis
|e₁) = (1 + i)î+ (1)ĵ+ (i)k, \e₂) = (i)î+ (3)ĵ+(1)k, |e3) = (0)î+ (28) ĵ + (0)k.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
