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Math Help - Linear equations

  1. #1
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    Linear equations

    Consider n linear equations
    a11* x1+ a12 * x2 ++ a1n * xn = b1
    a21* x1+ a22 * x2 ++ a2n * xn = b2
    .
    .
    .
    an1* x1+ an2 * x2 ++ ann * xn = bn
    There are several methods to asses and solve these equations . Finally by any method the solutions x1, x2 ,, xn are linear combinations of b1, b2 ,, bn . But
    is it possible without any algebraic (or simpler) calculations to show that x1, x2 ,, xn are such linear combinations?

    I am thinking about concepts which predict solutions without a direct effort
    to solve. That may be useful in some difficult situations ,of course no this simple example.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by mazaheri View Post
    Consider n linear equations
    a11* x1+ a12 * x2 ++ a1n * xn = b1
    a21* x1+ a22 * x2 ++ a2n * xn = b2
    .
    .
    .
    an1* x1+ an2 * x2 ++ ann * xn = bn
    There are several methods to asses and solve these equations . Finally by any method the solutions x1, x2 ,, xn are linear combinations of b1, b2 ,, bn . But
    is it possible without any algebraic (or simpler) calculations to show that x1, x2 ,, xn are such linear combinations?

    I am thinking about concepts which predict solutions without a direct effort
    to solve. That may be useful in some difficult situations ,of course no this simple example.
    This system may be expressed as
    A \bold{x} = \bold{b}
    where A is the coefficient matrix, x is the solution vector, and b is the constant vector in your equations.

    If the inverse of A exists, the solution is
    \bold{x} = A^{-1} \bold{b}
    which is easily shown to be linear in the b_i's by simply matrix multiplication.

    -Dan
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  3. #3
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    Joined
    Jul 2007
    Posts
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    Linear equations

    I apologize all.For simplicity the existance and uniquness of solution is the first assumption.Indeed matrix methods are a type of algebraic calculations .I mean no any such works.
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  4. #4
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    Linear equations

    At first I apologize all forgetting to notify that existence and uniqueness of answers has been supposed .Consider other equations with answers x'1, x'2 ,, x'n as
    following :

    a11* x'1+ a12 * x'2 ++ a1n * x'n = b'1
    a21* x'1+ a22 * x'2 ++ a2n * x'n = b'2
    .
    .
    .
    an1* x'1+ an2 * x'2 ++ ann * x'n = b'n



    Then regard the following equations :

    a11* x"1+ a12 * x"2 ++ a1n * x"n = C* b1 + D * b'1

    a21* x"1+ a22 * x"'2 ++ a2n * x"n = C* b2 + D * b'2
    .
    .
    .
    an1* x"1+ an2 * x"2 ++ ann * x"n = C* bn + D * b'n

    Uniqueness of answers shows that :
    x"i = C* xi + D * x'i , i =1 ,2,,n

    We can regard b1,b2,, bn as a function on X ={1,2,,n} such that :

    bi = f(i) , i=1,2,,n f: X ---- > C (complex values)

    X along its all subsets make a locally compact Hausdorff topological space. All functions X ---- > C are continuous by this topology . For each function such as f,there are x1,x2,.,xn that satisfy the linear equations . If we regard one of them ,e.g. xm ,xm is a linear functional of f. This functional is bounded and is defined on Cc(X).So according to Friesz theorem (real and complex analysis Walter Rudin , third edition ,19.6) ,there is a measure such as M (along some properties)that xm= int (fdM)on X . Clearly this integral can regard as a linear combination of b1,b2,, bn .
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