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Inner Product Properties

  • #1
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(Not an assigned problem...)
1. Homework Statement

pg 244 of "Mathematical Methods for Physics and Engineering" by Riley and Hobson says that given the following two properties of the inner product

pfor1.jpg


It follows that:
image.jpg


2. Attempt at a solution.
I think that both of these solutions are valid...but even if they are valid, is there a simpler and more intuitive way to derive these properties of inner products for a complex vector space from i and ii?
solution.png
 

Answers and Replies

  • #2
andrewkirk
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The item ##\langle \mathbf c^*\ |\ \lambda^*\mathbf a^*+\mu^*\mathbf b^*\rangle## in the second line of your attempt is undefined. Even if it had been defined, eg by assuming that the elements of the vector space were sequences of complex numbers, and defining the conjugate of the sequence to be the sequence of the conjugates, additional steps would still be needed to prove that second line. It does not follow automatically from the previous one.

Fortunately, you can fix the problem and shorten your proof by one line at the same time, by making the second line the result of applying Property 2 to line 1, and continuing on from there.
 
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