Linear Algebra · step 8 of 13

Rank, nullity & the rank–nullity theorem

The rank is the dimension of the image (column space); the nullity is the dimension of the kernel (null space). The rank–nullity theorem says rank + nullity = number of COLUMNS — that is, the dimension of the domain, not the codomain. Since both are read off the same row reduction as pivot and free-variable counts, the theorem is really a bookkeeping identity.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Row reduce onceEverything comes from this single computation.
Rank = number of pivotsEquivalently the number of non-zero rows in echelon form. Row rank equals column rank, which is a genuine theorem.
Nullity = number of free variablesColumns without pivots. Each contributes one basis vector to the kernel.
Apply the theorem to fill in the third quantityGiven any two of rank, nullity and column count, the theorem gives the third.

Worked example

A is a 3×5 matrix with rank 2. Find its nullity and say whether the associated map can be injective or surjective.

The map goes ℝ⁵ → ℝ³, so the domain has dimension 5.
Rank–nullity: 2 + nullity = 5, so nullity = 3.
Injective requires nullity 0, so it is not injective.
Surjective requires rank 3 (all of ℝ³), but the rank is 2.

Answer. Nullity 3; the map is neither injective nor surjective. Indeed any map ℝ⁵ → ℝ³ must have nullity ≥ 2 and so can never be injective.

Where marks get dropped

These are the specific errors that cost credit on rank, nullity & the rank–nullity theorem questions — QED's rubric penalises each of them separately.

Using the number of ROWS in the theorem. It is the number of columns — the dimension of the domain.
Computing the rank as the size of the matrix. Rank is at most the smaller dimension and is usually less.
Confusing the null space (in the domain) with the column space (in the codomain). They live in different spaces.

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Rank, nullity & the rank–nullity theorem — frequently asked questions

Why does row rank equal column rank?

Because row reduction preserves the row space and the dependence relations among columns, so the pivot count measures both. It is a theorem, not a definition.

What does full rank mean?

Rank equal to the smaller of the dimensions. For a square matrix, full rank is equivalent to invertibility and to a non-zero determinant.

How does this relate to solution sets?

The solution set of Ax = b, when non-empty, is a particular solution plus the null space — so its dimension is the nullity.

The rest of Linear Algebra

Matrices, systems, determinants, eigenvalues. Each subtopic below has its own method, worked example and mark-losing traps.

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