应用线性方程组理论证明矩阵秩的性质

Applying the Theory of System of Linear Equations to Prove the Property of Matrix Rank

利用矩阵秩的定义证明矩阵秩的性质时,需要使用行列式的性质,证明过程较为复杂。线性方程组解的理论与矩阵秩的内在联系,使得用线性方程组解的理论证明矩阵秩的性质成为可能。应用线性方程组解的理论,可将矩阵秩的等式证明转化为线性方程组解空间相等的证明;将矩阵秩的不等式的证明转化为解空间包含的证明。从行列式性质法的证明转化为集合间关系的证明,不仅简化了矩阵秩的性质的证明,而且证明过程便于理解。

When using the definition of the matrix rank to prove the property of the matrix rank,the property of the determinant needs to be applied,and the proof process is complex.The internal relationship between the theory of solution of system of linear equations and the rank of matrix implies that the property of the rank of matrix can be proved by the theory of solution of system of linear equations.Applying the theory of system of linear equations,the proof of equality for matrix rank is transformed into the proof of equality of solution spaces of system of linear equations,and the proof of inequality for matrix rank is transformed into the proof of inclusion in the solution space.The transformation from the proof of the determinant property method to the proof of the relationship between sets simplifies the proof of the property of the matrix rank and makes the proof more concise.