# Components and basis transform differently

• Ratzinger
In summary, the transformation of components and basis vectors under Lorentz transformation is different due to the nature of orthonormal coordinates and the effect of doubling a basis vector on the coordinate values. This is in contrast to orthogonal transformations where the components and basis are transformed by the same matrix.
Ratzinger
Why do components and basis vectors transform under Lorentz transformation differently (inverse Lorentz for basis), whereas for orthogonal transformation components and basis are transformed by same matrix?

Thank you

Are u talking about the group theoretical approach to relativity,or differential geometry ?I think you're mixing them.Components & basis vectors would imply geometry,while Lorentz transformation & orthogonal transformation would imply $\mbox{O(3)/SO(3)}$ or $SO(3,1)$...?

So which one would u prefer?

Daniel.

Ratzinger said:
Why do components and basis vectors transform under Lorentz transformation differently (inverse Lorentz for basis), whereas for orthogonal transformation components and basis are transformed by same matrix?

Thank you
I´m quite surprised that you want follow the dictatorship of relativism by tranforming even the basis. I expected you to rather conserve absolute values.

Well,that's the correct way to do it.Mathematcally.Physicists are rather sloppy,here,i must say,because are usually interested in how the commponents of (pseudo)tensors behave when a change of noninertial reference frames is done...They have no use for the transformation laws of the basis (co)vectors.

Daniel.

Ratzinger said:
Why do components and basis vectors transform under Lorentz transformation differently (inverse Lorentz for basis), whereas for orthogonal transformation components and basis are transformed by same matrix?

Thank you

The basis and the components ony transform in the same manner for a very special set of coordinates - orthonormal coordinates to be precise.

Probably the very simplest way to see why coordinates and vectors transform differently is to look at what happens when you double the length of the basis vector. It should be easy to see that if you double the basis vector, you halve the coordinate value, and vica-versa. That is why coordinates and basis vectors transform differently (oppositely).

Note that when you double one of the basis vectors, you are no longer in an orthonormal basis (you are orthogonal, but not normal).

So there is nothing special about the Lorentz transform - in general ALL coordinates transform differently than basis vectors. (Orthonormal coordinate systems are the exception - of course, they are a very common and important exception).

## 1. What is the difference between components and basis transformations?

Components refer to the individual parts or elements that make up a larger system, while basis transformations are mathematical operations that change the coordinate system used to represent a vector or matrix. In other words, components are the building blocks of a system, while basis transformations are a way to change the perspective from which we view those components.

## 2. How do components and basis transformations relate to each other?

Components can be represented in different coordinate systems, and basis transformations are used to convert between those systems. For example, a vector can have different components when represented in Cartesian coordinates versus polar coordinates. Basis transformations allow us to switch between these representations.

## 3. Why is it important to understand the difference between components and basis transformations?

Understanding the difference between components and basis transformations is crucial in many fields of science and engineering, particularly in physics and mathematics. It allows us to accurately describe and analyze complex systems and make predictions based on different perspectives or coordinate systems.

## 4. What are some common applications of basis transformations?

One common application of basis transformations is in quantum mechanics, where they are used to switch between different representations of quantum states. They are also used in computer graphics to rotate and scale images, and in signal processing to transform data into different domains for analysis.

There are many resources available to learn more about components and basis transformations, including textbooks, online tutorials, and courses. It is also helpful to practice applying these concepts to different problems and systems to deepen your understanding.

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