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Spaces of Matrices

smile

New member
Oct 15, 2013
19
Is anyone can help me to show The determinant function: det: M(n*n)-->R is continuous. and Justify it for the case n = 2.
where M(n*n) is a n by n matrix

Thanks
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,702
Is anyone can help me to show The determinant function: det: M(n*n)-->R is continuous. and Justify it for the case n = 2.
where M(n*n) is a n by n matrix

Thanks
What have you tried so far? I suggest you stick to the case $n=2$ and write down explicitly what it is that you need to prove the result in that case.
 

smile

New member
Oct 15, 2013
19
What have you tried so far? I suggest you stick to the case $n=2$ and write down explicitly what it is that you need to prove the result in that case.
Hello, i think we can take an open set in $R$ and show its pre-image is open, right? However, I do not know how to find the pre-image in this case.

Thanks
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,702
What I meant by writing it down explicitly is that if a $2\times2$ matrix has entries $w,\,x,\,y,\,z$ then its determinant is $wz-xy$. So your function $f:\mathbb{R}^4\to\mathbb{R}$ is $f(w,x,y,z) = wz-xy$. Why is that function continuous?

Hint: you know that sums and products of continuous functions are continuous. So all you really need to do here is to explain why the coordinate functions, such as $(w,x,y,z)\mapsto w$, are continuous.
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
I'd like to comment that the idea provided by Opalg, is based on the fact that all finite dimensional vector with the same dimension, are homeomorphic.
 

smile

New member
Oct 15, 2013
19
What I meant by writing it down explicitly is that if a $2\times2$ matrix has entries $w,\,x,\,y,\,z$ then its determinant is $wz-xy$. So your function $f:\mathbb{R}^4\to\mathbb{R}$ is $f(w,x,y,z) = wz-xy$. Why is that function continuous?

Hint: you know that sums and products of continuous functions are continuous. So all you really need to do here is to explain why the coordinate functions, such as $(w,x,y,z)\mapsto w$, are continuous.
Thanks for your help, I think I have figured out this problem.
:)

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I'd like to comment that the idea provided by Opalg, is based on the fact that all finite dimensional vector with the same dimension, are homeomorphic.
Thank you.:)