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a) Show that this definition of the inner product satisfies both of the above conditions.

a) Show that this definition of the inner product satisfies both of the above conditions.

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
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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. (a|Ab) = 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 C²: (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 (aļa) > 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. (a|Ab) = 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 C²: (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 (aļa) > 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.
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