Answered: (a/b) = a*b₁ + = (a²b₂ + ažb₁) + a₂b₂ 2

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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Question

The inner product, ⟨a|b⟩, of any two vectors a and b in a vector space must satisfy the conditions ⟨a|b⟩ = ⟨b|a⟩^∗ and ⟨a|λb⟩ = λ⟨a|b⟩, where λ is a scalar.

Consider the following candidate (provided) for the inner product for vectors in the two dimensional complex vectors space ℂ² where a = (a₁, a₂)^Tand b = (b₁, b₂)^T.

An additional requirement for the inner product is that ⟨a|a⟩ > 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 satisfy this additional positive-definite requirement.

Note: Consider re-writing ⟨a|a⟩ in terms of |a₁+ a₂|² and other positive terms.