# Rank and dimension

The rank of a matrix is the number of independent columns. The dimension of a space is the number of vectors in a basis. Rank plus nullity equals the number of columns.

Maths (university) · Part 09: Determinants, inverses and vector spaces · Card 8 of 9 · Last updated 2026-10-19

## Analogy

Rank counts how many truly different directions a matrix can produce.

## Example

**[[1,2,3],[2,4,6]]**

Row 2 is twice row 1.
Only 1 independent row, so rank = 1.
Columns: 3.
Nullity = 3 - 1 = 2.

## In practice

Rank tells you how many solutions a system has and how much information a matrix holds.

## Practice

### Flashcards

- **State the rank-nullity theorem.** Rank plus nullity equals the number of columns.

## Questions people ask

### What is the rank-nullity theorem?

For a matrix with n columns, the rank plus the nullity equals n. The nullity is the dimension of the set of solutions to A x equals 0. If the rank is 1 and there are 3 columns, then the nullity is 2.

### How do I find the rank of a matrix?

Do row operations until the matrix is in echelon form, then count the non-zero rows. For [[1,2],[2,4]], subtracting twice row 1 from row 2 gives a row of zeros, so only one non-zero row remains and the rank is 1.

## Related concepts

- [Linear transformation](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-9)
- [Inverse matrix](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-3)
- [Solving with the inverse](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-4)
- [Vector space](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-5)

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