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 $\Bbb R^2$?