Can a Matrix A Equalize Vectors u and v in Different Reference Frames?

In summary, the conversation discusses the concept of reference frames and their relationship to vectors in a vector space. It is mentioned that a matrix A can satisfy the equality u= A.v if it is represented by a non-singular n by n matrix whose columns are the coefficients of u when written as a linear combination of v. The issue of u and v being defined in different reference frames is also brought up.
  • #1
taybasta
2
0
Hi everyone,

Given two different reference frames in a vector space; say left and right. v is a vector defined in the left frame and u is a vector defined in the right frame.
What is the nature of a matrix A that can satisfy the equality u= A.v?
Thank you
 
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  • #2
If the vector space has dimension n, then A is represented by the n by n, non-singular, matrix whose columns are the coefficients of vectors in u when written as a linear combination of the vectors in v.
 
  • #3
Thanks HallsofIvy,
the problem is that the two vectors u and v are not defined in the same reference frame.
 

1. What is a coordinate system?

A coordinate system is a mathematical framework used to locate and describe points in space. It uses a set of numbers or coordinates to indicate the position of a point relative to a reference point or origin.

2. What are the different types of coordinate systems?

There are several types of coordinate systems, including Cartesian, polar, cylindrical, and spherical. Each type has its own unique way of representing points in space and is used for different purposes in various fields of science and engineering.

3. What is the importance of coordinate systems in science?

Coordinate systems are essential in science as they provide a standardized and consistent way of describing the location and movement of objects in space. They are used in various fields such as astronomy, physics, geology, and geography to analyze and interpret data.

4. How do you convert coordinates from one system to another?

To convert coordinates from one system to another, you need to use transformation equations specific to the two coordinate systems. These equations take into account the differences in the reference point, orientation, and unit of measurement between the two systems.

5. Can a single point have different coordinates in different coordinate systems?

Yes, a single point can have different coordinates in different coordinate systems. This is because each system has its own set of reference points and axes, resulting in different ways of representing the same point in space.

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