Inner product vs dot/scalar product

  • #1
cianfa72
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TL;DR Summary
About the difference between inner product and dot/scalar product
Hi,

from Penrose book "The Road to Reality" it seems to me inner product and dot/scalar product are actually different things.

Given a vector space ##V## an inner product ## \langle . | . \rangle## is defined between elements (i.e. vectors) of the vector space ##V## itself. Differently dot/scalar product ##\cdot## is defined between an element of the vector space ##V## and an element of the dual vector space ##V^*##.

Then given the inner product in ##V## there is a canonical isomorphism between ##V## and ##V^*## hence the result of the inner product between two vectors vs. the scalar product between the first vector and the dual vector canonically associated to the second vector is actually the same.

Does it make sense ? Thanks.
 
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  • #2
cianfa72 said:
TL;DR Summary: About the difference between inner product and dot/scalar product

from Penrose book "The Road to Reality" it seems to me inner product and dot/scalar product are actually different things.

Since your question is about terminology,
I suggest that you provide "quotes" of the definitions from the Penrose book.
 
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  • #3
Capture.JPG


Here Penrose defines the scalar product between a covector ##\alpha## and a vector ##\xi##.
 
  • #4
The same term can mean different things in different contexts. Scalar product can be used (and is used) to mean the same thing as inner product. It can also be used to mean the canonical pairing between vectors and covectors.
 
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  • #5

What is the difference between an inner product and a dot/scalar product?

An inner product is a generalization of the dot product in a vector space that does not necessarily have to be Euclidean. It can be defined on a complex vector space and has additional properties such as linearity in the first argument and conjugate symmetry. A dot product or scalar product is a specific type of inner product that is defined on Euclidean spaces and is commutative and distributive.

How are inner products and dot products calculated?

Inner products are calculated by taking the inner product of two vectors, which involves multiplying the components of the vectors element-wise and then summing the results. Dot products are a specific type of inner product that is calculated by multiplying the corresponding components of two vectors and then summing the results.

What are some applications of inner products and dot products?

Inner products and dot products are used in various fields such as physics, engineering, and computer science. They are used in vector spaces to define lengths, angles, and projections of vectors. In physics, they are used in quantum mechanics to represent physical states and calculate probabilities. In computer science, they are used in machine learning algorithms such as support vector machines.

Can inner products and dot products be negative?

Yes, both inner products and dot products can be negative. The sign of the inner product or dot product depends on the angle between the two vectors. If the angle is obtuse, the inner product or dot product will be negative. If the angle is acute, the inner product or dot product will be positive.

Are inner products and dot products the same in all vector spaces?

No, inner products and dot products are not the same in all vector spaces. Inner products are a generalization of dot products and can be defined on any vector space, including complex vector spaces. Dot products, on the other hand, are specific to Euclidean spaces and have additional properties such as commutativity and distributivity that may not hold in other vector spaces.

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