The inner product, (a|b), of any two vectors a and b in a vector space must satisfy the conditions 1. (alb) = (b|a)". %3D 2. (a|Ab) = X(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) + ažbz, where a = (as, az)" and b = (b, b;)". 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 (a|a) ensures that the norm of a vector is a real number. Show that the proposed definition of the inner product does satisty this additional positive-definite requirement. Hint: Consider re-writing (ala) in terms of |a, + az[² and other positive terms.
The inner product, (a|b), of any two vectors a and b in a vector space must satisfy the conditions 1. (alb) = (b|a)". %3D 2. (a|Ab) = X(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) + ažbz, where a = (as, az)" and b = (b, b;)". 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 (a|a) ensures that the norm of a vector is a real number. Show that the proposed definition of the inner product does satisty this additional positive-definite requirement. Hint: Consider re-writing (ala) in terms of |a, + az[² 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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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 fllowing candidate for the inner product for vectors in the two dimensional
complex vectors space C²:
(alb) = a;b, +(a;bz + ažbı) + ažbz,
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 (a|a) in terms of |a1 +azl² and other positive terms.
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