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

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

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.5: The Binomial Theorem

Problem 18E

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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.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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