# 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 = uFor 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)