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Math Help - Long Continuous proof

  1. #1
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    Long Continuous proof

    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.
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  2. #2
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    Re: Long Continuous proof

    Consider\ SO(2)\subset\ M_{2}(R),\ A\in SO(2) if\ detA = 1. Prove\ that\ SO(2)<br />
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
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  3. #3
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    Re: Long Continuous proof

    Why are you wrapping every thing is [tex] [/tex] ?
    Only put your equations in tex tags, not plain text.
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  4. #4
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    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?
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  5. #5
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    Re: Long Continuous proof

    Quote Originally Posted by maxgunn555 View Post
    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.
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  6. #6
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    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.
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  7. #7
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    Re: Long Continuous proof

    no help though?
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  8. #8
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    Re: Long Continuous proof

    the point is, det is a function from \mathbb{R}^4\to \mathbb{R}. you should already know that s(x,y) = x+y is continuous from \mathbb{R}^2\to \mathbb{R}, and that p(x,y) = xy is continuous from \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?
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