Vectors and Dual Vectors

In summary: One colloquial defining property of dual vectors is that they "eat vectors and produce a scalar." An example is -Eμdxμ = work done. But one can take any two regular vectors and form the sum FμGμ which would also be a scalar. So how are dual vectors different in this regard?Dual vectors are different because they are defined as linear maps from vectors to scalars, while regular vectors are simply elements of a vector space. So while you can form a scalar from two regular vectors by taking the dot product, you can also form a scalar from a dual vector and a regular vector by applying the
  • #36
I always thought, it would be a good idea to explain tensors in terms of bra and ket vectors. So I have tried to write it up. I wonder if this is helpful in explaining the above questions.
(it is work in progress, by the way)

My opinion is, that every measurement introduces a dual structure, since you need a quantity to tell you how much of something you got, and you need a link to reality, a unit. If you change your units, you will have to change the numbers in an inverse way. Say, you measure something to be 10m long. If you replace m with (2m), then you have 5 of (2m).
Now, for vectors you will have the same structure, if an object that you want to describe with vectors is invariant, and therefore describes an object in nature.
I think, for vector spaces one has 2 such structures. A vector itself can describe a distance in nature and hence an invariant object. That is why base vectors and the corresponding coefficients transform in an inverse way. To reflect that for the notation used, base vectors get lower indices and coefficients upper indices. So, if you combine this two objects, one with lower and one with upper indices, you get something invariant.
The length of a vector is invariant too, if it describes a length in nature. So the computation of a length has to produce a dual structure as well, or in other words, one structure with upper and one with lower indices, that will be combined. This leads to the introduction of dual vectors, that combine with vectors to create a scalar. This dual vectors are different from vectors, as much as coefficients are different from base vectors.
Going back to the first example, 10 * (m) = 5 * (2m). We could give the units a lower index and the coefficients an upper index as well.
Therefore, upper and lower indices are just a reminder of what things you can combine to create something invariant and therefore independent of an observer. To make a mathematical structure invariant, you will always have a combination of 2 structures that transform in an inverse way to each other.
 

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<h2>1. What is the difference between a vector and a dual vector?</h2><p>A vector is a mathematical object that represents magnitude and direction. It can be represented by an array of numbers or symbols. A dual vector, also known as a covector, is a mathematical object that represents a linear function on vectors. In other words, it is a mapping from vectors to scalars.</p><h2>2. How are vectors and dual vectors related?</h2><p>Vectors and dual vectors are related through a mathematical operation called the dot product. The dot product of a vector and a dual vector results in a scalar value. This relationship is often used in physics and engineering to calculate work, energy, and other physical quantities.</p><h2>3. Can vectors and dual vectors exist in different dimensions?</h2><p>Yes, vectors and dual vectors can exist in different dimensions. Vectors are typically represented as n-tuples, where n is the number of dimensions. Dual vectors, on the other hand, are represented as row vectors and can exist in a different number of dimensions than the corresponding vector.</p><h2>4. What is the geometric interpretation of a dual vector?</h2><p>The geometric interpretation of a dual vector is as a hyperplane in the vector space. In two dimensions, a dual vector can be thought of as a line perpendicular to the vector it is dual to. In three dimensions, a dual vector can be thought of as a plane perpendicular to the vector it is dual to.</p><h2>5. How are vectors and dual vectors used in machine learning?</h2><p>Vectors and dual vectors are used in machine learning to represent data and parameters. For example, vectors can be used to represent features of a dataset, while dual vectors can be used to represent the weights of a model. The dot product between a vector and a dual vector can be used to make predictions or update the model's parameters during training.</p>

1. What is the difference between a vector and a dual vector?

A vector is a mathematical object that represents magnitude and direction. It can be represented by an array of numbers or symbols. A dual vector, also known as a covector, is a mathematical object that represents a linear function on vectors. In other words, it is a mapping from vectors to scalars.

2. How are vectors and dual vectors related?

Vectors and dual vectors are related through a mathematical operation called the dot product. The dot product of a vector and a dual vector results in a scalar value. This relationship is often used in physics and engineering to calculate work, energy, and other physical quantities.

3. Can vectors and dual vectors exist in different dimensions?

Yes, vectors and dual vectors can exist in different dimensions. Vectors are typically represented as n-tuples, where n is the number of dimensions. Dual vectors, on the other hand, are represented as row vectors and can exist in a different number of dimensions than the corresponding vector.

4. What is the geometric interpretation of a dual vector?

The geometric interpretation of a dual vector is as a hyperplane in the vector space. In two dimensions, a dual vector can be thought of as a line perpendicular to the vector it is dual to. In three dimensions, a dual vector can be thought of as a plane perpendicular to the vector it is dual to.

5. How are vectors and dual vectors used in machine learning?

Vectors and dual vectors are used in machine learning to represent data and parameters. For example, vectors can be used to represent features of a dataset, while dual vectors can be used to represent the weights of a model. The dot product between a vector and a dual vector can be used to make predictions or update the model's parameters during training.

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