Prove Homeomorphism of {A in GL(n;R) | det(A)>0 & det(A)<0}

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In summary, the conversation discusses how to prove that the set of matrices with a positive determinant is homeomorphic to the set of matrices with a negative determinant. The participants suggest writing down a homeomorphism and considering the properties of the determinant and topology in GL(n;R) to better understand the concept.
  • #1
rifat
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Could anyone help me to prove the following question:
"How can we show that the set {A in GL(n;R) | det(A)>0} is homeomorphic to the set {A in GL(n;R) | det(A)<0}?"
 
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  • #2
Have you tried to write down a homeomorphism?
 
  • #3
Thanks for reply. Yah I mean Homeomorphism. Actually I would ike to know how can we show that the set {A in GL(n;R) | det(A)>0} is homeomorphic to the set {A in GL(n;R) | det(A)<0}. I didnt understand how can we explain the properties of Homeomorphism function here. Could you please tell me a little detail. Thanks again.
 
  • #4
I meant, have you tried to give an explicit homeomorphism between those two sets? It shouldn't be too hard.
 
  • #5
Rifat ,I hope this is not too obvious of a comment:
Think of the topology you are using in GL(n;R) , the properties of det (A),and it
should become clearer.
 

1. What is a homeomorphism?

A homeomorphism is a continuous function between two topological spaces that has a continuous inverse.

2. What is GL(n;R)?

GL(n;R) refers to the general linear group of n-dimensional matrices with real-valued entries. In other words, it is the set of all invertible n x n matrices with real numbers as entries.

3. How do you prove a homeomorphism?

To prove a homeomorphism, you must show that the function is continuous, bijective (both one-to-one and onto), and that its inverse is also continuous. In this case, you must also show that the matrices in set A have determinants greater than and less than 0, respectively.

4. Why is the condition det(A)>0 & det(A)<0 necessary for this proof?

This condition ensures that the function and its inverse are both well-defined and continuous. If either det(A)>0 or det(A)<0 is not satisfied, the inverse may not exist or may not be continuous, thus violating the requirements for a homeomorphism.

5. What are some real-world applications of homeomorphisms?

Homeomorphisms have various applications in fields such as topology, differential geometry, and physics. They are used to model and analyze complex systems, such as fluid flow or weather patterns, and to study the behavior of continuous functions in different spaces.

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