Geometric interpretation of a vector space and subspace?

Question

I understand how to manipulate vector spaces and subspaces and how to prove various statements about them, but I still don't fully understand what they represent geometrically.

I just need an intuitive grasp as to what these are. Is a vector space a geometric area that contains all possible vectors of the field $\Bbb F$? For example, if $V$ is a vector space over $\Bbb R^3$ then does that mean $V$ contains all vectors in three-dimensions that are part of $\Bbb R$? But then what is a subspace of $V$? Would that perhaps be a plane? Would it be a vector space in $\Bbb R^2$?

Answer 1

The classic example of a $3$-dimensional real vector space is the Euclidean space $\mathbb R^3$. Its $2$-dimensional subspaces are precisely all planes passing through the origin, and its $1$-dimensional subspaces are precisely all lines passing through the origin. The origin itself forms the singleton $\{0\}$ which is the only $0$-dimensional subspace.