Vector Space

A set $\mathbb{V}$ with an operation of addition, denoted $\vec{x}+\vec{y}$, and an operation of scalar multiplication, denoted $c\vec{x},c\in \mathbb{R}$ is called a vector space over $\mathbb{R}$ if for every $\vec{v},\vec{x},\vec{y}\in \mathbb{V}$ and for every $c,d\in \mathbb{R}$

$V1:\vec{x}+\vec{y}\in \mathbb{V}$ $V2:\vec{x}+\vec{y}=\vec{y}+\vec{x}$ $V3:(\vec{x}+\vec{y})+\vec{v}=\vec{x}+(\vec{y}+\vec{v})$ $V4:\text{There exists a vector }\vec{0}\in \mathbb{V},\text{ called the zero vector, so that }\vec{x}+\vec{0}=\vec{x}\text{ for every }\vec{x}\in \mathbb{V}$ $V5:\text{For every }\vec{x}\in \mathbb{V}\text{ there exists a }(-\vec{x})\in \mathbb{V}\text{ so that }\vec{x}+(-\vec{x})=\vec{0}$ $V6:c\vec{x}\in \mathbb{V}$ $V7:c(d\vec{x})=(cd)\vec{x}$ $V8:(c+d)\vec{x}=c\vec{x}+d\vec{x}$ $V9:c(\vec{x}+\vec{y})=c\vec{x}+c\vec{y}$ $V10:1\vec{x}=\vec{x}$

Difference between vector space and subspace?

A subspace is a vector space. It’s called a subspace when you’re talking about its relation to another space. By itself, it’s still a vector space.

So the difference is the dimension.

A proper subspace of a vector space is a vector space with a smaller dimension. An “improper” subspace would be a vector space of the same dimension.