Linear change of coordinates preserving a certain property

In summary, the linear change of coordinates preserving a certain property is a mathematical transformation that converts coordinates while maintaining a specific property, such as distance or angle measurements. Some common properties that can be preserved include distance, angle measurements, and orientation. This type of transformation is different from others because it can be described by a linear function and is often represented by a matrix. It has various applications in scientific research, particularly in fields such as physics, engineering, and computer graphics. Linear change of coordinates is closely related to linear algebra as it involves the use of matrices and vectors in its calculations.
  • #1
naturemath
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Doesn't a linear change of coordinates preserve complete intersection for a set of homogeneous polynomials, all of the same degree, in a polynomial ring?

That is, apply a change of coordinates to a set of homogeneous polynomials {f_1,... f_k} in C[x_1,...,x_M] to obtain {h_1,..., h_k}. Suppose now that the variety cut out by {h_1,...,h_k} is a complete intersection. Doesn't this imply that the original set of generators {f_1,... f_k} formed a complete intersection?

This seems very plausible.
 
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  • #2
Never mind. Please disregard this post.
 

Related to Linear change of coordinates preserving a certain property

1. What is the linear change of coordinates preserving a certain property?

The linear change of coordinates preserving a certain property refers to a mathematical transformation where the coordinates of a set of points in one coordinate system are converted to a new coordinate system while preserving a specific property, such as distance or angle measurements.

2. What are some common properties that can be preserved through linear change of coordinates?

Some common properties that can be preserved through linear change of coordinates include distance, angle measurements, and orientation. Other properties may also be preserved depending on the specific transformation being applied.

3. How is linear change of coordinates different from other types of coordinate transformations?

Linear change of coordinates is a specific type of coordinate transformation where the relationship between the old and new coordinate systems can be described by a linear function. This means that the transformation can be represented by a matrix and can be easily applied to a set of points.

4. What are the applications of linear change of coordinates in scientific research?

Linear change of coordinates has various applications in scientific research, particularly in fields such as physics, engineering, and computer graphics. It is used to simplify complex equations, transform data into a more usable form, and model physical systems.

5. How is linear change of coordinates related to linear algebra?

Linear change of coordinates is directly related to linear algebra as it involves the use of matrices and vectors to represent the transformation between coordinate systems. It also utilizes concepts such as linear transformations, determinants, and eigenvectors in its calculations.

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