Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
The inner product, (alb), of any two vectors a and b in a vector space must satisfy the conditions 1. (a|b) = (bla)". 2. (alAb) = A(a|b), where A is a scalar. Now consider the following candidate for the inner product for vectors in the two dimensional complex vectors space C?: (a|b) = a;b, +(a;bz + ažb;) + ażb», where a = (a1, az)" and b = (b, b2)". a) Show that this definition of the inner product satisfies both of the above conditions. b) Show that with this definition of the inner product, the pair of vectors a = (1, 0)" and b = (0, 1)" are not orthogonal, but the pair e = (1, 1)" and d = (1, – 1)" are orthogonal. c) An additional requirement for the inner product is that (ala) > 0 for all vectors a + 0. This positive-definite condition on (ala) ensures that the norm of a vector is a real number. Show that the proposed definition of the inner product does satisfy this additional positive-definite requirement. Hint: Consider re-writing (ala) in terms of |a, + a2l² and other positive terms.

The inner product, (alb), of any two vectors a and b in a vector space must satisfy the conditions 1. (a|b) = (bla)". 2. (alAb) = A(a|b), where A is a scalar. Now consider the following candidate for the inner product for vectors in the two dimensional complex vectors space C?: (a|b) = a;b, +(a;bz + ažb;) + ażb», where a = (a1, az)" and b = (b, b2)". a) Show that this definition of the inner product satisfies both of the above conditions. b) Show that with this definition of the inner product, the pair of vectors a = (1, 0)" and b = (0, 1)" are not orthogonal, but the pair e = (1, 1)" and d = (1, – 1)" are orthogonal. c) An additional requirement for the inner product is that (ala) > 0 for all vectors a + 0. This positive-definite condition on (ala) ensures that the norm of a vector is a real number. Show that the proposed definition of the inner product does satisfy this additional positive-definite requirement. Hint: Consider re-writing (ala) in terms of |a, + a2l² and other positive terms.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Do part b and c

The inner product, (alb), of any two vectors a and b in a vector space must satisfy the conditions 1. (a|b) = (b|a)". 2. (alAb) = (a|b), where A is a scalar. Now consider the following candidate for the inner product for vectors in the two dimensional complex vectors space C²: (a|b) = a;b; +(a;bz + ažbı) + ażbz, where a = (a1, az)" and b = (b, b2)". a) Show that this definition of the inner product satisfies both of the above conditions. b) Show that with this definition of the inner product, the pair of vectors a = (1, 0)" and b = (0, 1)" are not orthogonal, but the pair e = (1, 1)" and d = (1, – 1)* are orthogonal. c) An additional requirement for the inner product is that (a|a) > 0 for all vectors a + 0. This positive-definite condition on (ala) ensures that the norm of a vector is a real number. Show that the proposed definition of the inner product does satisfy this additional positive-definite requirement. Hint: Consider re-writing (ala) in terms of |a, + a2l² and other positive terms.
Transcribed Image Text:The inner product, (alb), of any two vectors a and b in a vector space must satisfy the conditions 1. (a|b) = (b|a)". 2. (alAb) = (a|b), where A is a scalar. Now consider the following candidate for the inner product for vectors in the two dimensional complex vectors space C²: (a|b) = a;b; +(a;bz + ažbı) + ażbz, where a = (a1, az)" and b = (b, b2)". a) Show that this definition of the inner product satisfies both of the above conditions. b) Show that with this definition of the inner product, the pair of vectors a = (1, 0)" and b = (0, 1)" are not orthogonal, but the pair e = (1, 1)" and d = (1, – 1)* are orthogonal. c) An additional requirement for the inner product is that (a|a) > 0 for all vectors a + 0. This positive-definite condition on (ala) ensures that the norm of a vector is a real number. Show that the proposed definition of the inner product does satisfy this additional positive-definite requirement. Hint: Consider re-writing (ala) in terms of |a, + a2l² and other positive terms.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS