Homeomorphism calculation help

Click For Summary
SUMMARY

The discussion centers on demonstrating that the set of matrices in GL(n;R) with a positive determinant is homeomorphic to the set of matrices in GL(n;R) with a negative determinant. A proposed method involves replacing the first row of a matrix A with its negative, effectively creating an explicit homeomorphism between the two sets. This transformation maintains the properties of the matrices while altering their determinants, confirming the homeomorphic relationship.

PREREQUISITES
  • Understanding of GL(n;R) and its properties
  • Knowledge of determinants and their significance in linear algebra
  • Familiarity with the concept of homeomorphism in topology
  • Basic matrix operations and transformations
NEXT STEPS
  • Study the properties of GL(n;R) in detail
  • Explore the implications of determinant signs in linear transformations
  • Research homeomorphism examples in topology
  • Examine matrix transformations and their effects on determinants
USEFUL FOR

Mathematicians, students of linear algebra, and anyone interested in topology and matrix theory will benefit from this discussion.

rifat
Messages
3
Reaction score
0
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}?"
 
Physics news on Phys.org
If you think of A as a matrix, how about just replacing the first row with the negative of itself? Isn't that an explicit homeomorphism?
 

Similar threads

Prove that a function from [0,1] to [0,1] is a homeomorphism
  • · Replies 5 ·
Replies
5
Views
2K
Prove that f is a homeomorphism iff g is continuous, fg=1 and gf=1
  • · Replies 5 ·
Replies
5
Views
2K
Is There a Simple Way to Show GL(n,C) is a Lie Group?
  • · Replies 4 ·
Replies
4
Views
2K
How Do Matrix ODEs Relate to Determinants and Traces?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad ##SO(3)## topology
  • · Replies 29 ·
Replies
29
Views
4K
Undergrad 2-sphere with any topology can't be homeomorphic to the plane
  • · Replies 5 ·
Replies
5
Views
2K
Proving ##(cof ~A)^t ~A = (det A)I##
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad ##SU(2)## homeomorphic with ##\mathbb S^3##
  • · Replies 61 ·
3
Replies
61
Views
7K
Undergrad Show that a "cross" is not a topological manifold
  • · Replies 20 ·
Replies
20
Views
4K
Homomorphisms with unknown groups
  • · Replies 14 ·
Replies
14
Views
2K