UNI_FOUNDATION · Linear algebra · LA.GAUSSIAN

Gaussian elimination

Gaussian elimination solves linear systems using elementary row operations.

Intuition

Understand first, compress later.

Each row operation replaces the system with an equivalent one having the same solution set but progressively simpler structure.

Switch representations

Matrix as a transformationA = [1 0; 0 1], det A = 1

original gridimage grid under Aimage of the unit square, area 1î ↦ (1, 0)ĵ ↦ (0, 1)

A sends î to (1, 0) and ĵ to (0, 1); the faint original grid becomes the gold image grid. The unit square (area 1) becomes the parallelogram of area |det A| = 1 — the size of the determinant is exactly that area factor.

Formal view

Allowed operations are row swaps, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. The target is row echelon form.

Why does it work?

Row operations correspond to reversible transformations of the equations and therefore preserve the solution set.