Hyperplane Intersections

Each equation $a_{i1} x_1 + \cdots + a_{in} x_n = b_i$ defines a hyperplane in $\mathbb{R}^n$ — a flat $(n-1)$-dimensional slice. Adding more equations means intersecting more hyperplanes.

When we add a new equation (a new row), one of three things happens: (1) the new row's vector is not in the span of the previous rows — then the solution space dimension drops by 1 (a genuine new constraint); (2) the new row is in the span of the previous rows and is consistent — the equation is redundant and the solution space is unchanged; (3) the new row is in the span of the previous rows but is inconsistent — the hyperplane misses the current solution set entirely (no solution).

This trichotomy exactly matches what we saw in Chapter 1 when studying two or three equations: the lines or planes either intersect transversally, overlap, or are parallel. Row rank gives us the rigorous version of that intuition for any number of equations in any dimension.