System of linear equations

In mathematics and linear algebra, a system of linear equations is a set of linear equations such as

Related Topics:
Mathematics - Linear algebra - Linear equation

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: 3x1 + 2x2 − x3 = 1

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: 2x1 − 2x2 + 4x3 = −2

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: −x1 + ½x2 − x3 = 0.

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The problem is to find those values for the unknowns x1, x2 and x3 which satisfy all three equations simultaneously.

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Systems of linear equations belong to the oldest problems in mathematics and they have many applications, such as in digital signal processing, estimation, forecasting and generally in linear programming and in the approximation of non-linear problems in numerical analysis. An efficient way to solve systems of linear equations is given by the Gauss-Jordan elimination or by the Cholesky decomposition.

Related Topics:
Digital signal processing - Linear programming - Numerical analysis - Gauss-Jordan elimination - Cholesky decomposition

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In general, a system with m linear equations and n unknowns can be written as

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: a11x1 + a12x2 + … + a1nxn = b1

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: a21x1 + a22x2 + … + a2nxn = b2

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:     :

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:     :

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: am1x1 + am2x2 + … + amnxn = bm,

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where x1, ... ,xn are the unknowns and the numbers aij are the coefficients of the system. We can separate the coefficients in a matrix as follows:

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:

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egin{bmatrix}

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a_{11} & a_{12} & cdots & a_{1n} \

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a_{21} & a_{22} & cdots & a_{2n} \

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dots & dots & ddots & dots \

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a_{m1} & a_{m2} & cdots & a_{mn} end{bmatrix}

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egin{bmatrix}

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x_1 \

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x_2 \

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dots \

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x_n

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end{bmatrix}

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