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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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