Gaussian Elimination Method

Linear Algebra for Machine Learning · Axiom Academy

The systematic method for solving systems through row reduction Gaussian elimination is an algorithm that transforms a system into upper triangular (echelon) form, making it easy to solve by back substitution. Partial pivoting: Choose the largest pivot to minimize numerical errors. After forward elimination, the system is in row echelon form: All zero rows (if any) are at the bottom First nonzero entry (pivot) in each row is to the right of pivots above All entries below pivots are zero Preservation: Elementary row operations don't change the solution set Simplicity: Upper triangular form is easy to solve with back substitution Universality: Works for any square or rectangular system Efficiency: O(n³) complexity—optimal for dense matrices

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