系数矩阵为行最简形的线性方程组的同解性

Identity of Solutions for Systems of Linear Equations with Row Reduced Echelon Coefficient Matrix

按照从简单到复杂的认知规律,用辅助方程组法,通过讨论最简单的线性方程组,证明了“系数矩阵为行最简形的同型同解线性方程组的增广矩阵相等”.与传统的线性相关性方法比较,本文的方法是构造性的.作为推论,证明了“线性方程组同解的充要条件是增广矩阵行等价”;简化了“矩阵的行最简形矩阵的唯一性”的证明.

According to the cognitive law from simple to complex,through discussing the simplest system of linear equations by means of the method of auxiliary system of equations,it is proved that if systems of linear equations of the same type with row reduced echelon coefficient matrix have the same solution,then their augmented matrix is equal.Compared with the method of traditional linear correlation,the method here is constructive.As consequences,it is proved that systems of linear equations have the same solution if and only if their augmented matrix is row equivalent,and the proof of the uniqueness of the row reduced echelon matrix for a matrix is simplified.