Morgan I.

asked • 02/11/18

Show that the set GL(n, R) of invertible n × n real matrices is open in the space of n × n matrices (metric spaces, Real analysis)

Show that the set GL(n, R) of invertible n × n real matrices is open in the space of n × n matrices, with the metric being the Euclidean metric in Rn^2 when a matrix is identified with a long, n2 -dimensional vector.
 
R stands for the Real Numbers, the Rn^2 where n is squared.
 
Someone explained this to me using determinant mapping, but I am unfamiliar with determinant mapping, and I am unsure that is the right way to prove this.
 

2 Answers By Expert Tutors

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Amaan M. answered • 02/17/18

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Christian S.

There's no need to include the condition that det is onto. A function f:X->Y,where X and Y are metric spaces (or any type of topological spaces), is continuous if and only if for every open set U in Y, the inverse image f^{-1}(U) is open in X.
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02/20/18

Christian S. answered • 02/20/18

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Award-winning math instructor with college ODE teaching experience

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