This row echelon form is the augmented matrix of a system of equations that is equivalent to the given system ( it has exactly the same solutions ).
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Some authors use the term Gaussian elimination to refer to the process until it has reached its upper triangular, or ( non-reduced ) row echelon form.
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Since elementary row operations preserve the row space of the matrix, the row space of the row echelon form is the same as that of the original matrix.
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My first ever class at university was about matrices and expressing systems of linear equations in terms of matrices equations, reducing to row echelon form and then solving.
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For matrices with integer coefficients, the Hermite normal form is a row echelon form that may be calculated using Euclidean division and without introducing any rational number or denominator.
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It works because the columns with pivots are a basis for the column space of the echelon form, and row reduction does not change the linear dependence relationships between the columns.
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Instead of stopping once the matrix is in echelon form, one could continue until the matrix is in " reduced " row echelon form, as it is done in the table.
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Instead of stopping once the matrix is in echelon form, one could continue until the matrix is in " reduced " row echelon form, as it is done in the table.
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The process of row reducing until the matrix is reduced is sometimes referred to as "'Gauss-Jordan elimination "', to distinguish it from stopping after reaching echelon form.
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Similarly, a system of equations is said to be in " reduced row echelon form " or in " canonical form " if its augmented matrix is in reduced row echelon form.
