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Definition of equivalent systems of linear equations

Created over 8 years ago, updated 25 days ago

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.)