Linear Algebra 35 | Rank-Nullity Theorem [dark version]

Rank–nullity theorem is the organizing rule behind how linear maps “trade” dimensions: for any linear map (equivalently, any matrix) from an n-dimensional input space, the dimension of its image plus the dimension of its kernel always equals n. In practical terms, if a map collapses some directions to zero, those lost dimensions show up exactly as extra freedom inside the kernel—so nothing about dimensional information is ever lost; it just gets redistributed.

The discussion begins by defining rank and nullity in geometric terms. For a matrix A, the rank is the dimension of its range (the set of all outputs the map can produce). Since the range sits inside the output space R^m, rank is bounded between 0 and min(n, m). A matrix is called “full rank” when its rank reaches this maximum possible value. Nullity is defined as the dimension of the kernel: the set of input vectors that get mapped to the zero vector. Nullity is also a nonnegative integer, and it measures how many independent directions in the input disappear under the map.

A concrete example makes the bookkeeping vivid. Consider a 2×3 matrix A. Because the output space is R^2, the rank can be at most 2. In the example, the first two columns already span R^2, so the rank is 2 and the range is the whole output space. That leaves one remaining input dimension (since the input space is R^3). A specific nonzero vector—(1, 1, 1)—is mapped to the zero vector because the three columns sum to zero. That vector spans the kernel, giving nullity 1. The dimensional accounting then reads rank + nullity = 2 + 1 = 3, matching the input dimension.

After the definitions, the theorem is proved using a dimension argument built from bases. Let k be the nullity, so the kernel has a basis of k vectors. By a basis extension result (the Steinitz exchange lemma), those k kernel vectors can be extended to a full basis of R^n by adding n−k additional vectors. When A is applied to this full basis, the kernel basis vectors contribute only zeros, so the image of the remaining n−k vectors spans the range. This gives one inequality: dim(range(A)) ≤ n−k.

To get equality, the proof shows the images of the added vectors are linearly independent. Any linear combination of the added vectors that maps to zero must lie in the kernel. But rewriting that combination in the full basis (kernel vectors plus added vectors) forces the coefficients on the added vectors to vanish, thanks to linear independence of the basis. Therefore the span of the images has dimension exactly n−k, meaning rank(A) = n−k. Rearranging yields rank(A) + nullity(A) = n, completing the proof.

The takeaway is operational: once rank or nullity is known, the other is determined immediately. The theorem is positioned as a core tool for later work, especially when analyzing solutions to systems of linear equations.