Linear Algebra
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The Rank-Nullity Theorem

The Rank-Nullity Theorem is one of the most important results in linear algebra: for any m×nm \times n matrix AA, Rank(A)+Nullity(A)=n\text{Rank}(A) + \text{Nullity}(A) = n The number of pivots plus the number of free variables always equals the total number of unknowns.

Geometrically: the domain Rn\mathbb{R}^n is split into two parts. The part that "goes to zero" (the null space, of dimension Nullity(AA)) and the part that "survives" and contributes to the output (of dimension Rank(AA)). These two dimensions must add up to nn.

A corollary: if AA is m×nm \times n, then Rank(AA) ≤min⁡(m,n)\leq \min(m, n). A 3×53 \times 5 matrix has rank at most 3; nullity at least 2.

Formal View

Theorem 3.17 (Rank-Nullity)
For any matrix A∈Rm×nA \in \mathbb{R}^{m \times n}:
Rank(A)+Nullity(A)=n\text{Rank}(A) + \text{Nullity}(A) = n
Equivalently, dim⁡(Col(A))+dim⁡(Null(A))=n=\dim(\text{Col}(A)) + \dim(\text{Null}(A)) = n = (number of columns).

This is sometimes called the "Fundamental Theorem of Linear Algebra" (first part).

Interactive Visualization

Rank-Nullity Theorem

Why This Matters

Rank-Nullity connects geometry (dimension) to computation (number of pivots and free variables) and provides fundamental constraints.

  • In control theory: Rank-Nullity tells you how many states are controllable vs. uncontrollable
  • In statistics: Rank-Nullity explains degrees of freedom in linear regression
  • Hyperplane corollary: any single linear equation aTx=0\mathbf{a}^T\mathbf{x} = 0 defines a subspace of dimension n−1n-1

Quiz

Question 1

Matrix AA is 5×85 \times 8 with rank 3. What is Nullity(AA)?

Question 2

A 3×33 \times 3 matrix with rank 3 has nullity 0.

Common Mistakes

  • Using mm (number of rows) instead of nn (number of columns) in the formula Rank + Nullity = nn.
  • Confusing Rank of AA (dim of column space) with number of rows mm — these can differ.