GENERAL SOLUTIONS OF A SYSTEM OF LINEAR EQUATIONS AND
Keywords:
Linear equations, general solution, Kronecker–Capelli theorem, matrix rank, Gauss elimination, affine space, free variables, linear independence, augmented matrix, particular solution, homogeneous solution, solution space structure.Abstract
This paper analyzes the general solutions of linear equation systems and the conditions for their existence. The study employs the Kronecker–Capelli theorem, matrix rank, and the Gauss elimination method as primary theoretical tools. The affine structure of the solution space and the relationship between the number of free variables and the dimension of the solution space are examined in detail. The results allow a comprehensive understanding of the solution structure from both algebraic and geometric perspectives.
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