Linear simulataneous equations, Guassian elimination and reduction to echelon form
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Linear simultaneous equations, matrices and Gaussian elimination
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To each system of simultaneous equations there corresponds a matrix form called the augmented matrix. This is best shown through examples. Thus, for example, corresponding to the simultaneous equation [Equation goes here - download the original to see it.] This can be extended to any number of simultaneous equations; for example, for 3 equations in 3 unknowns. To [Equation goes here - download the original to see it.] the augmented matrix is [Equation goes here - download the original to see it.] In general a system M of equations in n unknowns [Equation goes here - download the original to see it.] has augmented matrix [Equation goes here - download the original to see it.] A solution to a set of simultaneous equations is a set of equations each in one variable. For example [Equation goes here - download the original to see it.] is the solution to the first set of simultaneous equations. [Equation goes here - download the original to see it.] The solution could be written [Equation goes here - download the original to see it.] We call matrices of this type, where the 'square' part of the augment matrix has the form of an identity matrix, the row reduced form. The solution to a system of simultaneous equations corresponds to the row reduced form of the augmented matrix. We, therefore, seek legitimate operations on matrices that transform the allgmented matrix of a system of simultaneous equations to row reduced form. These legitimate row operations mimic the technique of elimination used to solve simultaneous equations. To show this, consider the solution, by elimination, of the first example. [Equation goes here - download the original to see it.] There the presence of the variables, x and y, are really non-essential; the process of elimination operates solely on the coefficients of the system of equations. Consequently, it makes sense to apply these operations to the allgmented matrix alone. Example [Equation goes here - download the original to see it.] Solution [Equation goes here - download the original to see it.] Then [Equation goes here - download the original to see it.] The row operations do to the rows of augmented matrix only what it is legitimate to do to the equations of a system of linear equations. to solve a system of linear equations we can add, subtract, multiply, divide and interchange the equations; similarly a row operation on a matrix is an operation of either (1) adding a multiple of one row to another (2) multiplying a row by a non-zero number (3) interchanging two rows. The technique of row reduction is called Gaussian elimination. to further illustrate this, we use this technique to display the solution to the second example. Example [Equation goes here - download the original to see it.] Solution [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.]
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Contents of
Linear simulataneous equations, Guassian elimination and reduction to echelon form
1 Linear simultaneous equations, matrices and Gaussian elimination
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