Equivalent bases for Vector Spaces over Complexes.

In summary, equivalent bases for vector spaces over the Complexes are defined as having the same orientation if there exists a matrix with a positive determinant that can transform one basis to another. However, in the case of a vector space over C, all bases may not be equivalent as the determinant of the transforming matrix may not be real-valued.
  • #1
WWGD
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Hi:

Given a fin.dim vector space V over R, and two different bases B_V,B_V'

for V , we say that B_V,B'_V are equivalent as bases ( or have the same

orientation) , if there exists a matrix T with TB=B', and DetT>0.

How do we define equivalent bases for vector spaces over the

Complexes.?. If W is a vector space over C, and we are given

bases B_W, B'_W , then the matrix L with LB=B' may not have

a real-valued determinant.

Is there then a way of defining equivalent bases in the second case.?

Thanks.
 
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  • #2
I think all bases are equivalent in a complex vector space because the the general linear group over the complex numbers is path connected.
 

1. What are equivalent bases for vector spaces over complexes?

Equivalent bases for vector spaces over complexes refer to sets of vectors that span the same space and can be used to represent any vector in that space. These bases may differ in terms of the individual vectors that make them up, but they ultimately have the same properties and can be used interchangeably.

2. How are equivalent bases for vector spaces over complexes determined?

Equivalent bases are determined by performing row and column operations on a given set of vectors. These operations include swapping, scaling, and adding or subtracting multiples of one vector to another. The resulting set of vectors will be equivalent to the original set and can be used as a basis for the same vector space.

3. Why are equivalent bases important in linear algebra?

Equivalent bases play a crucial role in linear algebra as they allow for more flexibility in representing vectors and performing calculations. They also help to simplify problems and make them more manageable by providing different ways to express the same information.

4. Can any set of vectors be an equivalent basis for a vector space over complexes?

No, not all sets of vectors can be equivalent bases for a vector space over complexes. The vectors in the set must be linearly independent, meaning that no vector in the set can be written as a linear combination of the other vectors. Additionally, the set must span the entire vector space, meaning that every vector in the space can be expressed as a linear combination of the basis vectors.

5. How are equivalent bases used in applications?

Equivalent bases are used in various applications in mathematics and science, including image and signal processing, quantum mechanics, and computer graphics. They are also essential in solving systems of linear equations and performing calculations in engineering and physics. In these fields, equivalent bases allow for efficient and accurate representations of complex systems and phenomena.

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