The geometry of linear systems

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.)

The geometric description assumes that each equation has at least one nonzero coefficient. An all-zero left side instead gives the whole ambient space when the constant is zero, or no solutions when it is nonzero.Copy