# Vector subspace

A vector subspace is a vector space that is a subset of another vector space. This means that all the properties of a vector space are satisfied. Let W be a non empty subset of a vector space V, then, W is a vector subspace if and only if the next 3 conditions are satisfied:[1][2]

1. additive identity – the element 0 is an element of W: 0 ∈ W
2. closed under addition – if x and y are elements of W, then x + y is also in W: x, yW implies x + yW
3. closed under scalar multiplication – if c is an element of a field K and x is in W, then cx is in W: cK and xW implies cxW.

If ${\displaystyle W_{1}}$ and ${\displaystyle W_{2}}$ are subspaces of a vector space ${\displaystyle V}$, then the sum and the direct sum of ${\displaystyle W_{1}}$ and ${\displaystyle W_{2}}$, denoted respectively by ${\displaystyle W_{1}+W_{2}}$ and ${\displaystyle W_{1}\oplus W_{2}}$,[3] are subspaces as well.[4]

## References

1. Axler, Sheldon (2015). Linear Algebra Done Right. Undergraduate Texts in Mathematics (Third ed.). Springer International Publishing. p. 18. doi:10.1007/978-3-319-11080-6. ISBN 978-3-319-11079-0.
2. "Subspace | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-23.
3. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-23.
4. "4.4: Sums and direct sum". Mathematics LibreTexts. 2013-11-07. Retrieved 2020-08-23.