Deriving Properties of Inner Products for Complex Vector Spaces

In summary, the conversation discusses properties of inner products in a complex vector space and the attempt at a solution involves deriving these properties from i and ii. The second line of the attempt is undefined and not automatically proven from the previous line. To fix this, the second line should be the result of applying Property 2 to the first line.
  • #1
DRose87
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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
 
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  • #2
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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1. What is an inner product?

An inner product is a mathematical operation that takes in two vectors and produces a scalar value. It is often used to measure the angle between two vectors or to project one vector onto another.

2. What are the properties of an inner product?

There are several properties that an inner product must satisfy, including linearity, symmetry, and positive definiteness. Linearity means that the inner product of a sum of vectors is equal to the sum of the inner products of each individual vector. Symmetry means that the order of the vectors does not matter. Positive definiteness means that the inner product of a vector with itself is always greater than or equal to zero.

3. How is an inner product different from a dot product?

An inner product is a more general concept, while a dot product is a specific type of inner product that is defined for vectors in Euclidean space. The dot product is also commutative, meaning the order of the vectors does not matter, while an inner product may not be commutative.

4. What is the geometric interpretation of an inner product?

The inner product between two vectors is equal to the product of their magnitudes and the cosine of the angle between them. This means that the geometric interpretation of an inner product is the projection of one vector onto the other multiplied by the length of the other vector.

5. How is an inner product used in applications?

An inner product is used in many areas of mathematics and science, such as in linear algebra, functional analysis, and quantum mechanics. It is also used in signal processing, data compression, and machine learning algorithms.

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