Rank-Nullity Theorem

The rank-nullity theorem splits the domain of a linear map into what survives (column space) and what collapses to zero (null space).

1. The idea

The rank of $A$ is the dimension of its column space (the space of outputs it can produce), and the nullity is the dimension of its null space ${x : A x = 0}$ (inputs it destroys).

The rank-nullity theorem states that for $A \in R^{m \times n}$ , rank plus nullity equals $n$ , the number of columns. This ties together how much information a linear map preserves versus loses.

Rank also equals the number of nonzero singular values or eigenvalues (for a diagonalizable matrix), and it determines whether a system $A x = b$ has a unique solution, infinitely many, or none.

2. Why it matters

Rank deficiency in a covariance matrix (e.g. more assets than time periods) signals collinearity, and interviewers use rank arguments to check whether a system of constraints is solvable.

3. Core mathematics

Rank-nullity theorem

rank (A) + nullity (A) = n

Full rank (square)

rank (A) = n ⟺ det (A) \neq = 0

Null space

N (A) = {x : A x = 0}

4. How to recognize it

Think of this when…

Questions about whether a system has a unique solution
Collinear assets or redundant constraints in a portfolio problem
'How many free parameters remain' type phrasing

5. Worked example

Worked example

Let $A = (122436)$ . Find rank and nullity.

1
Row 2 is twice row 1, so rank is 1.
$rank (A) = 1$
2
Apply rank-nullity with $n = 3$ columns.
$nullity (A) = 3 - 1 = 2$
Answer: rank = 1, nullity = 2

6. Common mistakes

Applying rank-nullity with $m$ (rows) instead of $n$ (columns)
Assuming a matrix with no zero rows must be full rank
Forgetting that rank is bounded by $min (m, n)$

7. Quick check

For $A$ a $3 \times 5$ matrix with rank 3, the nullity is:

A square matrix with nontrivial null space has determinant:

8. Practice

2 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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