Applying the Theory of System of Linear Equations to Prove the Property of Matrix Rank
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 rela-tionship 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 e-quations.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 be-tween sets simplifies the proof of the property of the matrix rank and makes the proof more concise.
solution of linear equationsrank of matrixdimension of linear space
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