# Long Continuous proof

• Dec 2nd 2011, 02:51 PM
maxgunn555
Long Continuous proof
$\displaystyle The\ space\ M_{2}R\ of\ all\ 2x2\ matrices\ over\ R\ can\ be\ identified\ with\ R^4: \left(\begin{array}{cc}A_{11}&A_{12}\\A_{21}&A_{22 }\end{array}\right)\rightarrow\left(\begin{array}{ cc}A_{11}\\A_{12}\\A_{21}\\A_{22}\end{array}\right ) Therefore\ we\ can\ introduce\ on\ M_{2}R\ the\ Euclidean\ metric:\parallel\left(\begin{array}{cc}A_{11}&A_{1 2}\\A_{21}&A_{22}\end{array}\right)\parallel\ =(\mid\ A_{11}-B_{11}\mid^2\ +\mid\ A_{12}-B_{12}\mid^2\ +\mid\ A_{21}-B_{21}\mid^2\ +\mid\ A_{22}-B_{22}\mid^2)^{1/2}\\ in\ particular\ we\ have\ the\ notion\ of\ a\ corresponding\ mapping\ f:\ M_{2}R\rightarrow\ R.\ Clearly\ f\ corresponds\ to\ a\ mapping\ f~\ :\ R^4\rightarrow\ R,\\ f(\left(\begin{array}{cc}A_{11}&A_{12}\\A_{21}&A_{ 22}\end{array}\right))=f~(A_{11}, A_{12},A_{21}, A_{22})\\ Prove\ that\ M_{2}R\rightarrow\ R,a\rightarrow\ det(a)\ is\ continuous$

btw all the large As and Bs should actually be small and the a ---> det(a) should actually be large; A----> det(A). Any help is valued thnx.
• Dec 2nd 2011, 03:12 PM
maxgunn555
Re: Long Continuous proof
$\displaystyle Consider\ SO(2)\subset\ M_{2}(R),\ A\in SO(2) if\ detA = 1. Prove\ that\ SO(2) is\ a\ closed\ subset\ in\ M_{2}(R)$

there's another part to the question. analysis really is the hardest module. the uni blackboard is really quite a mess of 100s pages just for this module, i just need any help to get me in the right direction thanks :)
• Dec 2nd 2011, 03:14 PM
Plato
Re: Long Continuous proof
Why are you wrapping every thing is  ?
Only put your equations in tex tags, not plain text.
• Dec 2nd 2011, 03:56 PM
maxgunn555
Re: Long Continuous proof
for me i think it'll be quicker eventually. why does it not read well? any advice for the question though?
• Dec 2nd 2011, 04:16 PM
Plato
Re: Long Continuous proof
Quote:

Originally Posted by maxgunn555
why does it not read well? any advice for the question though?

Well, I for one will bother to read through it.
I started to edit it but gave up.
I doubt many will be willing to deal with it.
• Dec 3rd 2011, 02:51 PM
maxgunn555
Re: Long Continuous proof
yeah you're right, you can't have letters written like they're maths letters its look weird and is quicker not to.
• Dec 4th 2011, 12:22 PM
maxgunn555
Re: Long Continuous proof
no help though?
• Dec 5th 2011, 04:40 AM
Deveno
Re: Long Continuous proof
the point is, det is a function from $\displaystyle \mathbb{R}^4\to \mathbb{R}$. you should already know that s(x,y) = x+y is continuous from $\displaystyle \mathbb{R}^2\to \mathbb{R}$, and that p(x,y) = xy is continuous from $\displaystyle \mathbb{R}^2\to \mathbb{R}$.

so det(a,b,c,d) = s(p(a,d),-p(b,c)), which is a composition of continuous maps, yes?