# What is a vector space?

A vector space is just a set V of vectors on which are defined two operations: **addition **and **multiplication**. Note: They don't actually have to be addition and multiplication, they just have to hold for these ten axioms, and addition and multiplication hold.

- The sum of vectors
**u**and**v,**denoted**u + v**is in V (closure under addition) - u + v = v + u
- (u + v) + w = u + (v + w)
- There is a
**zero**vector 0 (doesn't actually have to be 0) in V such that u + 0 = u - For each u in V, there is a vector -u in V such that u + (-u) = 0
- The scalar multiple of
**u**by c, denoted by**c**u is in V. (closure under scalar multiplication) - c(
**u + v) =**c**u +**c**v** - (c + d)
**u =**c**u**+ d**u** - c(d
**u) =**cd**u** - 1u = u

Basically, showing that it's closed under addition and scalar multiplication makes it a vector space given our normal + and *, because of the properties of real numbers. When the operations aren't orthodox, you must show all 10.

# What is a subspace?

A **subspace** is a subset of a vector space that has three properties

- It has the zero vector
- It's closed under vector addition (Axiom 1 from before)
- It's closed under scalar multiplication (Axiom 6 from before)

So, a subspace **is **a vector space. Conversely, every vector space is a subspace too! (of itself or larger spaces)