Rank-Nullity Theorem

Linear Algebra (Matrices) · Axiom Academy

LESSON The Rank-Nullity Theorem Every input dimension is either crushed to zero or carried into the output — and the two counts always add up to n . 1. Every Input Dimension Has Two Fates Feed all n dimensions of the domain into T . Watch each one sort itself into exactly one of two bins: the directions T crushes to (the kernel, counted by the nullity ) and the directions that survive into the output (the image, counted by the rank ). The full bar is n ; the split is the theorem. nullity (kernel) + rank (image) = dimension of the domain — reachable outputs (the column space); . 2. Read the Split Straight Off the RREF You never have to guess the split — row-reduce A and the columns sort themselves. Pivot columns count the rank; free columns (no pivot) count the nullity. Every column is one or the other, so they must sum to the total number of columns, which is n . Watch a matrix's four columns light up as the row-reduction lands. A leading 1 in the RREF marks a linearly independent output direction. Here columns 1 and 3 pivot, so . No pivot means a free variable — a direction the map collapses. Columns 2 and 4 are free, so . 3. Grow the Kernel, Shrink the Image Because the two counts are locked to a fixed total n , they trade off perfectly. Slide the nullity up from 0 and the rank slides down by exactly the same amount — the two bars are two ends of one budget of length n . Here n=3 : at every step the kernel and image bars together fill the whole domain.

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