# Span and basis

The span of vectors is everything you can build from them by scaling and adding. A basis is an independent set that spans the whole space.

Maths (university) · Part 09: Determinants, inverses and vector spaces · Card 7 of 9 · Last updated 2026-10-19

## Analogy

A basis is a minimal set of paint colours that can mix every shade you need.

## Example

**Basis (1,0) and (0,1)**

Any (2,3) = 2(1,0) + 3(0,1).
Coordinates are 2 and 3.
Another basis: (1,1) and (1,-1).
Then (2,3) = 2.5(1,1) - 0.5(1,-1).

## In practice

A basis gives every vector in the space a unique set of coordinates.

## Practice

### Flashcards

- **What is a basis?** An independent set of vectors that spans the whole space.

### Interview questions

**Explain linear independence and test whether (1,2) and (2,4) are independent.**

- Independent if the only combination giving zero is trivial
- (2,4) = 2(1,2)
- So 2(1,2) - (2,4) = 0 with non-zero coefficients
- Hence dependent

## Questions people ask

### Is a basis for a space unique?

No. Many different bases exist for the same space. In the plane, (1,0) and (0,1) form one, and (1,1) and (1,-1) form another. What stays fixed is the number of vectors in any basis, which is the dimension of the space.

### How do I find coordinates in a new basis?

Write the vector as a combination of the basis vectors and solve for the coefficients. For (2,3) with basis (1,1) and (1,-1), solve a plus b equals 2 and a minus b equals 3. This gives a equals 2.5 and b equals -0.5.

## Related concepts

- [Linear independence](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-6)
- [Vector space](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-5)
- [Rank and dimension](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-8)
- [Linear transformation](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-9)

Web page: https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-7
Study this card in the app: https://studyvisually.com/app/learn?category=maths-university&card=maths-university-part-9-concept-7