# What is a "subspace" in linear algebra?

A subset (call it W) of vectors is a subspace when it satisfies these conditions:

- W contains the zero vector
- If two vectors u and v are in W, then u+v must also be in W. (This is called "closed under addition")
- If the vector v is in W, and k is some scalar (ie just some number), then kv must also be in W. (This is called "closed under scalar multiplication").
- In other words, every linear combination of two vectors in W is also in W.

Testing these conditions is the best way to see if W is a subspace.

Some common subspaces of $ {\mathbb R}^3 $

- The zero vector, $ \vec 0 $
- A line running through the origin
- A plane passing through the origin
- $ {\mathbb R}^3 $