# Vector space

A set of vectors you can add together and scale by numbers, while always staying inside the set. A subspace is a vector space sitting inside a bigger one.

Maths (university) · Part 09: Determinants, inverses and vector spaces · Card 5 of 9 · Last updated 2026-10-19

## Analogy

A flat sheet of paper extending forever: any two arrows on it add to another arrow on it.

## Example

**Lines in the plane**

The line y = 2x is a subspace.
(1,2) + (2,4) = (3,6), still on it.
The line y = 2x + 1 is not.
It misses the origin (0,0).

## In practice

Gives one framework covering arrows, polynomials, functions and data.

## Practice

### Flashcards

- **Why must a subspace contain the zero vector?** Scaling any member by 0 gives the zero vector.

## Questions people ask

### Why must a subspace contain the zero vector?

Scaling any vector in the subspace by zero must stay inside it, and that gives the zero vector. So a line that misses the origin, like y equals 2x plus 1, cannot be a subspace. This is a quick first test. Always check the zero vector before testing anything else.

### Are polynomials a vector space?

Yes. Adding two polynomials gives a polynomial, and multiplying by a number gives one too. So polynomials of degree at most 2 form a vector space with 3 dimensions, built from 1, x and x squared. Functions and sequences can also form vector spaces.

## Related concepts

- [Linear independence](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-6)
- [Span and basis](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-7)
- [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)

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