# Vector Space

A set $V$ with an operation of addition, denoted $x+y $, and an operation of scalar multiplication, denoted $cx,c∈R$ is called a vector space over $R$ if for every $v,x,y ∈V$ and for every $c,d∈R$

$V1:x+y ∈V$ $V2:x+y =y +x$ $V3:(x+y )+v=x+(y +v)$ $V4:There exists a vector0∈V,called the zero vector, so thatx+0=xfor everyx∈V$ $V5:For everyx∈Vthere exists a(−x)∈Vso thatx+(−x)=0$ $V6:cx∈V$ $V7:c(dx)=(cd)x$ $V8:(c+d)x=cx+dx$ $V9:c(x+y )=cx+cy $ $V10:1x=x$

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.