The mathematical foundation of system solving rests on linear algebra principles, particularly the theory of linear transformations and vector spaces.
Matrix Representation Theory:
Every system of linear equations can be represented as Ax = b, where A is the coefficient matrix, x is the variable vector, and b is the constant vector. The solution existence depends on the relationship between rank(A) and rank([A|b]).
The fundamental theorem states: If rank(A) = rank([A|b]) = n (number of variables), the system has a unique solution. If rank(A) = rank([A|b]) < n, infinite solutions exist. If rank(A) ≠ rank([A|b]), no solution exists.
Gaussian Elimination Algorithm:
The algorithm performs elementary row operations to transform the augmented matrix into row echelon form. This process has O(n³) complexity for an n×n system, making it efficient for moderate-sized systems.
Computational Considerations:
Partial pivoting prevents numerical instability by avoiding division by small numbers. For large systems, specialized methods like LU decomposition or iterative techniques may be preferred.
The condition number of matrix A determines numerical stability - well-conditioned matrices (low condition number) provide reliable solutions, while ill-conditioned matrices may amplify rounding errors.