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Math Help - Prove a binary operation is closed

  1. #1
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    Prove a binary operation is closed

    If a_1, a_2,b_1, b_2 \in \mathbb{R} , A_1,A_2 \in T where A_1=\left(\begin{array}{cc}a_1&-b_1\\b_1&a_1\end{array}\right) and A_2=\left(\begin{array}{cc}a_2&-b_2\\b_2&a_2\end{array}\right). Is A_1+A_2 necessarily a close operation on T?
    Last edited by novice; August 6th 2010 at 06:25 PM.
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  2. #2
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    Well, what do you think? When you add two matrices like this, is the result a matrix of the same form?
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  3. #3
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    The proof in my book says, "Since A_1,A_2 \in T, then A_1+A_2 \in T. Therefor the operation is closed on T." I have very difficulties accepting it because somewhere else in the book says, "If T \subset S and  a,b \in T, a*b is closed on S, but need not be closed on T.
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  4. #4
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    Book example

    Let T=\{\left[\begin{array}{cc}a&-b\\b&a\end{array}\right]: a,b \in \mathbb{R}\}. Is T closed under matrix addition.

    Proof:
    Let A_1, A_2 \in T. Then A_1=\left[\begin{array}{cc}a_1&-b_1\\b_1&a_1\end{array}\right] and A_2=\left[\begin{array}{cc}a_2&-b_2\\b_2&a_2\end{array}\right] for some  a_1,b_1,a_2,b_2 \in \mathbb{R}. Then
    A_1+A_2=\left[\begin{array}{cc}a_1+a_2&-(b_1+b_2)\\b_1+b_2&a_1+a_2\end{array}\right]. Since A_1+A_2\in T, it follows that T is closed under addition.

    We know that A_1+A_2 \in \mathbb{R}, but how could it be possible that A_1+A_2 \in T ?

    Oh, yeah, never mind. I see now that if A_1+A_2 \in \mathbb{R} then A_1+A_2 \in T according to T=\{\left[\begin{array}{cc}a&-b\\b&a\end{array}\right]: a,b \in \mathbb{R}\}
    Thanks again sir.
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  5. #5
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    That looks right to me.
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