Definition of equivalent systems of linear equations

The geometry of linear systems connects algebraic equations with geometric objects.

Each linear equation in $n$ variables represents a hyperplane in $\mathbb{R}^n$ — a flat geometric object of dimension $n-1$:

• In $\mathbb{R}^2$: each equation is a line
• In $\mathbb{R}^3$: each equation is a plane
• In $\mathbb{R}^n$: each equation is a hyperplane

Solving a system of equations geometrically means finding the intersection of these hyperplanes. The intersection of flat objects is itself flat, which explains why the solution set of a linear system is always:

• Empty (parallel hyperplanes that never meet)
• A single point (hyperplanes meet at exactly one point)
• An affine subspace of dimension $\geq 1$ (hyperplanes overlap along a line, plane, etc.)