Spaces of Matrices

[smile]

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]

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]

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]

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]

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]

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.